Abstract group 7801596.a: $C_{199}:C_{198}^2$

• Label: 7801596.a
• Order: \( 2^{2} \cdot 3^{4} \cdot 11^{2} \cdot 199 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{4} \cdot 11^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \cdot 11 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{5} \cdot 5 \cdot 11^{2} \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 11 \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$C_{199}:C_{198}^2$
Order:	\(7801596\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 11^{2} \cdot 199 \)
Exponent:	\(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Automorphism group:	Group of order \(468095760\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 5 \cdot 11^{2} \cdot 199 \)
Composition factors:	$C_2$ x 2, $C_3$ x 4, $C_{11}$ x 2, $C_{199}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	6	9	11	18	22	33	66	99	198	199	398	597	1194	1791	2189	3582	4378	6567	13134	19701	39402
Elements	1	399	1196	4380	13140	21900	41796	69660	187080	569160	1707480	5146200	198	198	396	396	1188	1980	1188	1980	3960	3960	11880	11880	7801596
Conjugacy classes	1	3	8	24	72	120	216	360	960	2880	8640	25920	1	1	2	2	6	10	6	10	20	20	60	60	39402
Divisions	1	3	4	12	12	12	36	36	48	144	144	432	1	1	1	1	1	1	1	1	1	1	1	1	896

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b \mid a^{198}=b^{39402}=1, b^{a}=b^{7129} \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 133 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_2$ $\, \times\, $ $C_9$ $\, \times\, $ $C_{11}$ $\, \times\, $ $F_{199}$
Semidirect product:	$C_{199}$ $\,\rtimes\,$ $C_{198}^2$	$(C_{4378}:C_{22})$ $\,\rtimes\,$ $C_9^2$	$(C_{1791}:C_9)$ $\,\rtimes\,$ $C_{22}^2$	$(C_{19701}:C_{99})$ $\,\rtimes\,$ $C_2^2$	all 6
Trans. wreath product:	not computed
Possibly split product:	$F_{199}$ . $C_{198}$	$C_{198}$ . $F_{199}$	$C_{39402}$ . $C_{198}$	$(C_{99}\times F_{199})$ . $C_2$	all 163

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{198}^{2} \simeq C_{2}^{2} \times C_{9}^{2} \times C_{11}^{2}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 1622 normal subgroups (66 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{198}$	$G/Z \simeq$ $F_{199}$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_{198}^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_{66}\times F_{199}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{39402}$	$G/\operatorname{Fit} \simeq$ $C_{198}$
Radical:	$R \simeq$ $C_{199}:C_{198}^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{13134}$	$G/\operatorname{soc} \simeq$ $C_3\times C_{198}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$
199-Sylow subgroup:	$P_{ 199 } \simeq$ $C_{199}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 9 (order at least 866844) are shown, as well as all normal subgroups of any index.

Order 7801596: $C_{199}:C_{198}^2$

Order 3900798: $C_{99}\times F_{199}$ x 2, $C_{19701}:C_{198}$

Order 2600532: $C_{66}\times F_{199}$ x 3, $C_{39402}:C_{66}$

Order 1950399: $C_{19701}:C_{99}$

Order 1300266: $C_{33}\times F_{199}$ x 6, $C_{13134}:C_{99}$ x 3, $C_{19701}:C_{66}$ x 2, $C_{19701}:C_{66}$

Order 866844: $C_{22}\times F_{199}$ x 9, $C_{4378}:C_{198}$ x 2, $C_{1791}:C_{22}^2$, $C_{13134}:C_{66}$

Order 709236: $C_{18}\times F_{199}$ x 11, $C_{2189}:C_{18}^2$

Order 650133: $C_{6567}:C_{99}$ x 3, $C_{19701}:C_{33}$

Order 433422: $C_{11}\times F_{199}$ x 18, $C_{2189}:C_{198}$ x 9, $C_{2189}:C_{198}$ x 4, $C_{2189}:C_{198}$ x 2, $C_{6567}:C_{66}$ x 2, $C_{19701}:C_{22}$ x 2, $C_{13134}:C_{33}$, $C_{3582}:C_{11}^2$

Order 354618: $C_9\times F_{199}$ x 22, $C_{1791}:C_{198}$ x 11, $C_{1791}:C_{198}$ x 2, $C_{4378}:C_9^2$

Order 288948: $C_{4378}:C_{66}$ x 3, $C_{13134}:C_{22}$

Order 236412: $C_6\times F_{199}$ x 33, $C_{3582}:C_{66}$ x 11, $C_{1194}:C_{198}$ x 3, $C_{3582}.C_{66}$

Order 216711: $C_{2189}:C_{99}$ x 9, $C_{2189}:C_{99}$ x 2, $C_{19701}:C_{11}$, $C_{6567}:C_{33}$

Order 177309: $C_{1791}:C_{99}$ x 11, $C_{1791}:C_{99}$

Order 144474: $C_{2189}:C_{66}$ x 6, $C_{2189}:C_{66}$ x 3, $C_{2189}:C_{66}$ x 2, $C_{13134}:C_{11}$

Order 118206: $C_3\times F_{199}$ x 66, $C_{1194}:C_{99}$ x 33, $C_{1791}:C_{66}$ x 22, $C_{1791}:C_{66}$ x 11, $C_{597}:C_{198}$ x 6, $C_{1194}:C_{99}$ x 3, $C_{1791}:C_{66}$ x 2, $C_{1791}.C_{66}$

Order 96316: $C_{4378}:C_{22}$

Order 78804: $C_2\times F_{199}$ x 99, $C_{398}:C_{198}$ x 22, $C_{1194}:C_{66}$ x 11, $C_{3582}:C_{22}$ x 11, $C_{398}:C_{198}$ x 9, $C_{398}:C_{198}$ x 2, $C_{1194}:C_{66}$, $C_{99}\times D_{398}$

Order 72237: $C_{2189}:C_{33}$ x 3, $C_{2189}:C_{33}$

Order 64476: $C_{3582}:C_{18}$

Order 59103: $C_{597}:C_{99}$ x 33, $C_{1791}:C_{33}$ x 11, $C_{597}:C_{99}$ x 3, $C_{1791}:C_{33}$

Order 48158: $C_{2189}:C_{22}$ x 2, $C_{2189}:C_{22}$

Order 39402: $F_{199}$ x 198, $C_{199}:C_{198}$ x 99, $C_{199}:C_{198}$ x 44, $C_{1791}:C_{22}$ x 22, $C_{199}:C_{198}$ x 22, $C_{597}:C_{66}$ x 22, $C_{199}:C_{198}$ x 18, $C_{1791}:C_{22}$ x 11, $C_{1194}:C_{33}$ x 11, $C_{199}:C_{198}$ x 9, $C_{199}:C_{198}$ x 4, $C_{99}\times D_{199}$ x 2, $C_{199}:C_{198}$ x 2, $C_{597}:C_{66}$ x 2, $C_{66}\times C_{199}:C_3$, $C_{39402}$

Order 39204: $C_{198}^2$

Order 32238: $C_{1791}:C_{18}$ x 2, $C_{1791}:C_{18}$

Order 26268: $C_{398}:C_{66}$ x 33, $C_{1194}:C_{22}$ x 11, $C_{398}:C_{66}$ x 3, $C_{66}\times D_{199}$

Order 24079: $C_{2189}:C_{11}$

Order 21492: $C_{1194}:C_{18}$ x 3, $C_{3582}:C_6$

Order 19701: $C_{199}:C_{99}$ x 99, $C_{199}:C_{99}$ x 22, $C_{597}:C_{33}$ x 11, $C_{1791}:C_{11}$ x 11, $C_{199}:C_{99}$ x 9, $C_{199}:C_{99}$ x 2, $C_{597}:C_{33}$, $C_{19701}$

Order 16119: $C_{1791}:C_9$

Order 13134: $C_{199}:C_{66}$ x 66, $C_{199}:C_{66}$ x 33, $C_{199}:C_{66}$ x 22, $C_{1194}:C_{11}$ x 11, $C_{199}:C_{66}$ x 6, $C_{199}:C_{66}$ x 3, $C_{33}\times D_{199}$ x 2, $C_{13134}$

Order 10746: $C_{597}:C_{18}$ x 6, $C_{1194}:C_9$ x 3, $C_{1791}:C_6$ x 2, $C_{1791}:C_6$

Order 8756: $C_{398}:C_{22}$ x 11, $C_{11}\times D_{398}$

Order 7164: $C_{398}:C_{18}$ x 9, $C_{398}:C_{18}$ x 2, $C_{1194}:C_6$, $C_9\times D_{398}$

Order 6567: $C_{199}:C_{33}$ x 33, $C_{199}:C_{33}$ x 11, $C_{199}:C_{33}$ x 3, $C_{6567}$

Order 5373: $C_{597}:C_9$ x 3, $C_{1791}:C_3$

Order 4378: $C_{199}:C_{22}$ x 22, $C_{199}:C_{22}$ x 11, $C_{11}\times D_{199}$ x 2, $C_{4378}$

Order 3582: $C_{199}:C_{18}$ x 18, $C_{199}:C_{18}$ x 9, $C_{199}:C_{18}$ x 4, $C_{199}:C_{18}$ x 2, $C_{597}:C_6$ x 2, $C_9\times D_{199}$ x 2, $C_{1194}:C_3$, $C_{3582}$

Order 2388: $C_{398}:C_6$ x 3, $C_3\times D_{398}$

Order 2189: $C_{199}:C_{11}$ x 11, $C_{2189}$

Order 1791: $C_{199}:C_9$ x 9, $C_{199}:C_9$ x 2, $C_{597}:C_3$, $C_{1791}$

Order 1194: $C_{199}:C_6$ x 6, $C_{199}:C_6$ x 3, $C_3\times D_{199}$ x 2, $C_{1194}$

Order 796: $D_{398}$

Order 597: $C_{199}:C_3$ x 3, $C_{597}$

Order 398: $D_{199}$ x 2, $C_{398}$

Order 199: $C_{199}$

Order 198: $C_{198}$

Order 121: $C_{11}^2$

Order 99: $C_{99}$

Order 81: $C_9^2$

Order 66: $C_{66}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 18: $C_{18}$

Order 11: $C_{11}$

Order 9: $C_9$

Order 6: $C_6$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 9 (order at least 866844) are shown, as well as all normal subgroups of any index.

Order 7801596: $C_{199}:C_{198}^2$

Order 3900798: $C_{19701}:C_{198}$, $C_{99}\times F_{199}$

Order 2600532: $C_{39402}:C_{66}$, $C_{66}\times F_{199}$

Order 1950399: $C_{19701}:C_{99}$

Order 1300266: $C_{19701}:C_{66}$, $C_{19701}:C_{66}$, $C_{13134}:C_{99}$, $C_{33}\times F_{199}$

Order 866844: $C_{4378}:C_{198}$, $C_{1791}:C_{22}^2$, $C_{13134}:C_{66}$, $C_{22}\times F_{199}$

Order 709236: $C_{2189}:C_{18}^2$, $C_{18}\times F_{199}$

Order 650133: $C_{19701}:C_{33}$, $C_{6567}:C_{99}$

Order 433422: $C_{2189}:C_{198}$, $C_{6567}:C_{66}$, $C_{19701}:C_{22}$, $C_{2189}:C_{198}$, $C_{13134}:C_{33}$, $C_{3582}:C_{11}^2$, $C_{11}\times F_{199}$, $C_{2189}:C_{198}$

Order 354618: $C_{1791}:C_{198}$, $C_{4378}:C_9^2$, $C_9\times F_{199}$, $C_{1791}:C_{198}$

Order 288948: $C_{13134}:C_{22}$, $C_{4378}:C_{66}$

Order 236412: $C_{3582}.C_{66}$, $C_{3582}:C_{66}$, $C_{1194}:C_{198}$, $C_6\times F_{199}$

Order 216711: $C_{19701}:C_{11}$, $C_{2189}:C_{99}$, $C_{6567}:C_{33}$, $C_{2189}:C_{99}$

Order 177309: $C_{1791}:C_{99}$, $C_{1791}:C_{99}$

Order 144474: $C_{13134}:C_{11}$, $C_{2189}:C_{66}$, $C_{2189}:C_{66}$, $C_{2189}:C_{66}$

Order 118206: $C_{1791}:C_{66}$, $C_{1791}.C_{66}$, $C_{1791}:C_{66}$, $C_{1791}:C_{66}$, $C_{1194}:C_{99}$, $C_{597}:C_{198}$, $C_3\times F_{199}$, $C_{1194}:C_{99}$

Order 96316: $C_{4378}:C_{22}$

Order 78804: $C_{1194}:C_{66}$, $C_{1194}:C_{66}$, $C_{398}:C_{198}$, $C_{3582}:C_{22}$, $C_{398}:C_{198}$, $C_{99}\times D_{398}$, $C_{398}:C_{198}$, $C_2\times F_{199}$

Order 72237: $C_{2189}:C_{33}$, $C_{2189}:C_{33}$

Order 64476: $C_{3582}:C_{18}$

Order 59103: $C_{597}:C_{99}$, $C_{1791}:C_{33}$, $C_{1791}:C_{33}$, $C_{597}:C_{99}$

Order 48158: $C_{2189}:C_{22}$, $C_{2189}:C_{22}$

Order 39402: $C_{66}\times C_{199}:C_3$, $C_{99}\times D_{199}$, $C_{39402}$, $C_{199}:C_{198}$, $C_{199}:C_{198}$, $C_{597}:C_{66}$, $C_{199}:C_{198}$, $C_{1791}:C_{22}$, $C_{1791}:C_{22}$, $C_{199}:C_{198}$, $C_{199}:C_{198}$, $C_{199}:C_{198}$, $F_{199}$, $C_{1194}:C_{33}$, $C_{597}:C_{66}$, $C_{199}:C_{198}$

Order 39204: $C_{198}^2$

Order 32238: $C_{1791}:C_{18}$, $C_{1791}:C_{18}$

Order 26268: $C_{66}\times D_{199}$, $C_{398}:C_{66}$, $C_{1194}:C_{22}$, $C_{398}:C_{66}$

Order 24079: $C_{2189}:C_{11}$

Order 21492: $C_{3582}:C_6$, $C_{1194}:C_{18}$

Order 19701: $C_{597}:C_{33}$, $C_{199}:C_{99}$, $C_{199}:C_{99}$, $C_{597}:C_{33}$, $C_{1791}:C_{11}$, $C_{199}:C_{99}$, $C_{199}:C_{99}$, $C_{19701}$

Order 16119: $C_{1791}:C_9$

Order 13134: $C_{199}:C_{66}$, $C_{199}:C_{66}$, $C_{199}:C_{66}$, $C_{33}\times D_{199}$, $C_{199}:C_{66}$, $C_{199}:C_{66}$, $C_{1194}:C_{11}$, $C_{13134}$

Order 10746: $C_{1791}:C_6$, $C_{1791}:C_6$, $C_{1194}:C_9$, $C_{597}:C_{18}$

Order 8756: $C_{11}\times D_{398}$, $C_{398}:C_{22}$

Order 7164: $C_{1194}:C_6$, $C_9\times D_{398}$, $C_{398}:C_{18}$, $C_{398}:C_{18}$

Order 6567: $C_{6567}$, $C_{199}:C_{33}$, $C_{199}:C_{33}$, $C_{199}:C_{33}$

Order 5373: $C_{1791}:C_3$, $C_{597}:C_9$

Order 4378: $C_{4378}$, $C_{11}\times D_{199}$, $C_{199}:C_{22}$, $C_{199}:C_{22}$

Order 3582: $C_{1194}:C_3$, $C_{199}:C_{18}$, $C_{199}:C_{18}$, $C_{597}:C_6$, $C_9\times D_{199}$, $C_{199}:C_{18}$, $C_{199}:C_{18}$, $C_{3582}$

Order 2388: $C_3\times D_{398}$, $C_{398}:C_6$

Order 2189: $C_{2189}$, $C_{199}:C_{11}$

Order 1791: $C_{597}:C_3$, $C_{199}:C_9$, $C_{1791}$, $C_{199}:C_9$

Order 1194: $C_{1194}$, $C_3\times D_{199}$, $C_{199}:C_6$, $C_{199}:C_6$

Order 796: $D_{398}$

Order 597: $C_{597}$, $C_{199}:C_3$

Order 398: $C_{398}$, $D_{199}$

Order 199: $C_{199}$

Order 198: $C_{198}$

Order 121: $C_{11}^2$

Order 99: $C_{99}$

Order 81: $C_9^2$

Order 66: $C_{66}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 18: $C_{18}$

Order 11: $C_{11}$

Order 9: $C_9$

Order 6: $C_6$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7801596: $C_{199}:C_{198}^2$ ($C_1$)

Order 3900798: $C_{99}\times F_{199}$ ($C_2$) x 2, $C_{19701}:C_{198}$ ($C_2$)

Order 2600532: $C_{66}\times F_{199}$ ($C_3$) x 3, $C_{39402}:C_{66}$ ($C_3$)

Order 1950399: $C_{19701}:C_{99}$ ($C_2^2$)

Order 1300266: $C_{33}\times F_{199}$ ($C_6$) x 6, $C_{13134}:C_{99}$ ($C_6$) x 3, $C_{19701}:C_{66}$ ($C_6$) x 2, $C_{19701}:C_{66}$ ($C_6$)

Order 866844: $C_{22}\times F_{199}$ ($C_9$) x 9, $C_{4378}:C_{198}$ ($C_9$) x 2, $C_{1791}:C_{22}^2$ ($C_9$), $C_{13134}:C_{66}$ ($C_3^2$)

Order 709236: $C_{18}\times F_{199}$ ($C_{11}$) x 11, $C_{2189}:C_{18}^2$ ($C_{11}$)

Order 650133: $C_{6567}:C_{99}$ ($C_2\times C_6$) x 3, $C_{19701}:C_{33}$ ($C_2\times C_6$)

Order 433422: $C_{11}\times F_{199}$ ($C_{18}$) x 18, $C_{2189}:C_{198}$ ($C_{18}$) x 9, $C_{2189}:C_{198}$ ($C_{18}$) x 4, $C_{2189}:C_{198}$ ($C_{18}$) x 2, $C_{6567}:C_{66}$ ($C_3\times C_6$) x 2, $C_{19701}:C_{22}$ ($C_{18}$) x 2, $C_{13134}:C_{33}$ ($C_3\times C_6$), $C_{3582}:C_{11}^2$ ($C_{18}$)

Order 354618: $C_9\times F_{199}$ ($C_{22}$) x 22, $C_{1791}:C_{198}$ ($C_{22}$) x 11, $C_{1791}:C_{198}$ ($C_{22}$) x 2, $C_{4378}:C_9^2$ ($C_{22}$)

Order 288948: $C_{4378}:C_{66}$ ($C_3\times C_9$) x 3, $C_{13134}:C_{22}$ ($C_3\times C_9$)

Order 236412: $C_6\times F_{199}$ ($C_{33}$) x 33, $C_{3582}:C_{66}$ ($C_{33}$) x 11, $C_{1194}:C_{198}$ ($C_{33}$) x 3, $C_{3582}.C_{66}$ ($C_{33}$)

Order 216711: $C_{2189}:C_{99}$ ($C_2\times C_{18}$) x 9, $C_{2189}:C_{99}$ ($C_2\times C_{18}$) x 2, $C_{19701}:C_{11}$ ($C_2\times C_{18}$), $C_{6567}:C_{33}$ ($C_6^2$)

Order 177309: $C_{1791}:C_{99}$ ($C_2\times C_{22}$) x 11, $C_{1791}:C_{99}$ ($C_2\times C_{22}$)

Order 144474: $C_{2189}:C_{66}$ ($C_3\times C_{18}$) x 6, $C_{2189}:C_{66}$ ($C_3\times C_{18}$) x 3, $C_{2189}:C_{66}$ ($C_3\times C_{18}$) x 2, $C_{13134}:C_{11}$ ($C_3\times C_{18}$)

Order 118206: $C_3\times F_{199}$ ($C_{66}$) x 66, $C_{1194}:C_{99}$ ($C_{66}$) x 33, $C_{1791}:C_{66}$ ($C_{66}$) x 22, $C_{1791}:C_{66}$ ($C_{66}$) x 11, $C_{597}:C_{198}$ ($C_{66}$) x 6, $C_{1194}:C_{99}$ ($C_{66}$) x 3, $C_{1791}:C_{66}$ ($C_{66}$) x 2, $C_{1791}.C_{66}$ ($C_{66}$)

Order 96316: $C_{4378}:C_{22}$ ($C_9^2$)

Order 78804: $C_2\times F_{199}$ ($C_{99}$) x 99, $C_{398}:C_{198}$ ($C_{99}$) x 22, $C_{1194}:C_{66}$ ($C_3\times C_{33}$) x 11, $C_{3582}:C_{22}$ ($C_{99}$) x 11, $C_{398}:C_{198}$ ($C_{99}$) x 9, $C_{398}:C_{198}$ ($C_{99}$) x 2, $C_{1194}:C_{66}$ ($C_3\times C_{33}$), $C_{99}\times D_{398}$ ($C_{99}$)

Order 72237: $C_{2189}:C_{33}$ ($C_6\times C_{18}$) x 3, $C_{2189}:C_{33}$ ($C_6\times C_{18}$)

Order 64476: $C_{3582}:C_{18}$ ($C_{11}^2$)

Order 59103: $C_{597}:C_{99}$ ($C_2\times C_{66}$) x 33, $C_{1791}:C_{33}$ ($C_2\times C_{66}$) x 11, $C_{597}:C_{99}$ ($C_2\times C_{66}$) x 3, $C_{1791}:C_{33}$ ($C_2\times C_{66}$)

Order 48158: $C_{2189}:C_{22}$ ($C_9\times C_{18}$) x 2, $C_{2189}:C_{22}$ ($C_9\times C_{18}$)

Order 39402: $F_{199}$ ($C_{198}$) x 198, $C_{199}:C_{198}$ ($C_{198}$) x 99, $C_{199}:C_{198}$ ($C_{198}$) x 44, $C_{1791}:C_{22}$ ($C_{198}$) x 22, $C_{199}:C_{198}$ ($C_{198}$) x 22, $C_{597}:C_{66}$ ($C_3\times C_{66}$) x 22, $C_{199}:C_{198}$ ($C_{198}$) x 18, $C_{1791}:C_{22}$ ($C_{198}$) x 11, $C_{1194}:C_{33}$ ($C_3\times C_{66}$) x 11, $C_{199}:C_{198}$ ($C_{198}$) x 9, $C_{199}:C_{198}$ ($C_{198}$) x 4, $C_{99}\times D_{199}$ ($C_{198}$) x 2, $C_{199}:C_{198}$ ($C_{198}$) x 2, $C_{597}:C_{66}$ ($C_3\times C_{66}$) x 2, $C_{66}\times C_{199}:C_3$ ($C_3\times C_{66}$), $C_{39402}$ ($C_{198}$)

Order 32238: $C_{1791}:C_{18}$ ($C_{11}\times C_{22}$) x 2, $C_{1791}:C_{18}$ ($C_{11}\times C_{22}$)

Order 26268: $C_{398}:C_{66}$ ($C_3\times C_{99}$) x 33, $C_{1194}:C_{22}$ ($C_3\times C_{99}$) x 11, $C_{398}:C_{66}$ ($C_3\times C_{99}$) x 3, $C_{66}\times D_{199}$ ($C_3\times C_{99}$)

Order 24079: $C_{2189}:C_{11}$ ($C_{18}^2$)

Order 21492: $C_{1194}:C_{18}$ ($C_{11}\times C_{33}$) x 3, $C_{3582}:C_6$ ($C_{11}\times C_{33}$)

Order 19701: $C_{199}:C_{99}$ ($C_2\times C_{198}$) x 99, $C_{199}:C_{99}$ ($C_2\times C_{198}$) x 22, $C_{597}:C_{33}$ ($C_6\times C_{66}$) x 11, $C_{1791}:C_{11}$ ($C_2\times C_{198}$) x 11, $C_{199}:C_{99}$ ($C_2\times C_{198}$) x 9, $C_{199}:C_{99}$ ($C_2\times C_{198}$) x 2, $C_{597}:C_{33}$ ($C_6\times C_{66}$), $C_{19701}$ ($C_2\times C_{198}$)

Order 16119: $C_{1791}:C_9$ ($C_{22}^2$)

Order 13134: $C_{199}:C_{66}$ ($C_3\times C_{198}$) x 66, $C_{199}:C_{66}$ ($C_3\times C_{198}$) x 33, $C_{199}:C_{66}$ ($C_3\times C_{198}$) x 22, $C_{1194}:C_{11}$ ($C_3\times C_{198}$) x 11, $C_{199}:C_{66}$ ($C_3\times C_{198}$) x 6, $C_{199}:C_{66}$ ($C_3\times C_{198}$) x 3, $C_{33}\times D_{199}$ ($C_3\times C_{198}$) x 2, $C_{13134}$ ($C_3\times C_{198}$)

Order 10746: $C_{597}:C_{18}$ ($C_{11}\times C_{66}$) x 6, $C_{1194}:C_9$ ($C_{11}\times C_{66}$) x 3, $C_{1791}:C_6$ ($C_{11}\times C_{66}$) x 2, $C_{1791}:C_6$ ($C_{11}\times C_{66}$)

Order 8756: $C_{398}:C_{22}$ ($C_9\times C_{99}$) x 11, $C_{11}\times D_{398}$ ($C_9\times C_{99}$)

Order 7164: $C_{398}:C_{18}$ ($C_{11}\times C_{99}$) x 9, $C_{398}:C_{18}$ ($C_{11}\times C_{99}$) x 2, $C_{1194}:C_6$ ($C_{33}^2$), $C_9\times D_{398}$ ($C_{11}\times C_{99}$)

Order 6567: $C_{199}:C_{33}$ ($C_6\times C_{198}$) x 33, $C_{199}:C_{33}$ ($C_6\times C_{198}$) x 11, $C_{199}:C_{33}$ ($C_6\times C_{198}$) x 3, $C_{6567}$ ($C_6\times C_{198}$)

Order 5373: $C_{597}:C_9$ ($C_{22}\times C_{66}$) x 3, $C_{1791}:C_3$ ($C_{22}\times C_{66}$)

Order 4378: $C_{199}:C_{22}$ ($C_9\times C_{198}$) x 22, $C_{199}:C_{22}$ ($C_9\times C_{198}$) x 11, $C_{11}\times D_{199}$ ($C_9\times C_{198}$) x 2, $C_{4378}$ ($C_9\times C_{198}$)

Order 3582: $C_{199}:C_{18}$ ($C_{11}\times C_{198}$) x 18, $C_{199}:C_{18}$ ($C_{11}\times C_{198}$) x 9, $C_{199}:C_{18}$ ($C_{11}\times C_{198}$) x 4, $C_{199}:C_{18}$ ($C_{11}\times C_{198}$) x 2, $C_{597}:C_6$ ($C_{33}\times C_{66}$) x 2, $C_9\times D_{199}$ ($C_{11}\times C_{198}$) x 2, $C_{1194}:C_3$ ($C_{33}\times C_{66}$), $C_{3582}$ ($C_{11}\times C_{198}$)

Order 2388: $C_{398}:C_6$ ($C_{33}\times C_{99}$) x 3, $C_3\times D_{398}$ ($C_{33}\times C_{99}$)

Order 2189: $C_{199}:C_{11}$ ($C_{18}\times C_{198}$) x 11, $C_{2189}$ ($C_{18}\times C_{198}$)

Order 1791: $C_{199}:C_9$ ($C_{22}\times C_{198}$) x 9, $C_{199}:C_9$ ($C_{22}\times C_{198}$) x 2, $C_{597}:C_3$ ($C_{66}^2$), $C_{1791}$ ($C_{22}\times C_{198}$)

Order 1194: $C_{199}:C_6$ ($C_{33}\times C_{198}$) x 6, $C_{199}:C_6$ ($C_{33}\times C_{198}$) x 3, $C_3\times D_{199}$ ($C_{33}\times C_{198}$) x 2, $C_{1194}$ ($C_{33}\times C_{198}$)

Order 796: $D_{398}$ ($C_{99}^2$)

Order 597: $C_{199}:C_3$ ($C_{66}\times C_{198}$) x 3, $C_{597}$ ($C_{66}\times C_{198}$)

Order 398: $D_{199}$ ($C_{99}\times C_{198}$) x 2, $C_{398}$ ($C_{99}\times C_{198}$)

Order 199: $C_{199}$ ($C_{198}^2$)

Order 198: $C_{198}$ ($F_{199}$)

Order 99: $C_{99}$ ($C_2\times F_{199}$)

Order 66: $C_{66}$ ($C_3\times F_{199}$)

Order 33: $C_{33}$ ($C_6\times F_{199}$)

Order 22: $C_{22}$ ($C_9\times F_{199}$)

Order 18: $C_{18}$ ($C_{11}\times F_{199}$)

Order 11: $C_{11}$ ($C_{18}\times F_{199}$)

Order 9: $C_9$ ($C_{22}\times F_{199}$)

Order 6: $C_6$ ($C_{33}\times F_{199}$)

Order 3: $C_3$ ($C_{66}\times F_{199}$)

Order 2: $C_2$ ($C_{99}\times F_{199}$)

Order 1: $C_1$ ($C_{199}:C_{198}^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7801596: $C_{199}:C_{198}^2$ ($C_1$)

Order 3900798: $C_{19701}:C_{198}$ ($C_2$), $C_{99}\times F_{199}$ ($C_2$)

Order 2600532: $C_{39402}:C_{66}$ ($C_3$), $C_{66}\times F_{199}$ ($C_3$)

Order 1950399: $C_{19701}:C_{99}$ ($C_2^2$)

Order 1300266: $C_{19701}:C_{66}$ ($C_6$), $C_{19701}:C_{66}$ ($C_6$), $C_{13134}:C_{99}$ ($C_6$), $C_{33}\times F_{199}$ ($C_6$)

Order 866844: $C_{4378}:C_{198}$ ($C_9$), $C_{1791}:C_{22}^2$ ($C_9$), $C_{13134}:C_{66}$ ($C_3^2$), $C_{22}\times F_{199}$ ($C_9$)

Order 709236: $C_{2189}:C_{18}^2$ ($C_{11}$), $C_{18}\times F_{199}$ ($C_{11}$)

Order 650133: $C_{19701}:C_{33}$ ($C_2\times C_6$), $C_{6567}:C_{99}$ ($C_2\times C_6$)

Order 433422: $C_{2189}:C_{198}$ ($C_{18}$), $C_{6567}:C_{66}$ ($C_3\times C_6$), $C_{19701}:C_{22}$ ($C_{18}$), $C_{2189}:C_{198}$ ($C_{18}$), $C_{13134}:C_{33}$ ($C_3\times C_6$), $C_{3582}:C_{11}^2$ ($C_{18}$), $C_{11}\times F_{199}$ ($C_{18}$), $C_{2189}:C_{198}$ ($C_{18}$)

Order 354618: $C_{1791}:C_{198}$ ($C_{22}$), $C_{4378}:C_9^2$ ($C_{22}$), $C_9\times F_{199}$ ($C_{22}$), $C_{1791}:C_{198}$ ($C_{22}$)

Order 288948: $C_{13134}:C_{22}$ ($C_3\times C_9$), $C_{4378}:C_{66}$ ($C_3\times C_9$)

Order 236412: $C_{3582}.C_{66}$ ($C_{33}$), $C_{3582}:C_{66}$ ($C_{33}$), $C_{1194}:C_{198}$ ($C_{33}$), $C_6\times F_{199}$ ($C_{33}$)

Order 216711: $C_{19701}:C_{11}$ ($C_2\times C_{18}$), $C_{2189}:C_{99}$ ($C_2\times C_{18}$), $C_{6567}:C_{33}$ ($C_6^2$), $C_{2189}:C_{99}$ ($C_2\times C_{18}$)

Order 177309: $C_{1791}:C_{99}$ ($C_2\times C_{22}$), $C_{1791}:C_{99}$ ($C_2\times C_{22}$)

Order 144474: $C_{13134}:C_{11}$ ($C_3\times C_{18}$), $C_{2189}:C_{66}$ ($C_3\times C_{18}$), $C_{2189}:C_{66}$ ($C_3\times C_{18}$), $C_{2189}:C_{66}$ ($C_3\times C_{18}$)

Order 118206: $C_{1791}:C_{66}$ ($C_{66}$), $C_{1791}.C_{66}$ ($C_{66}$), $C_{1791}:C_{66}$ ($C_{66}$), $C_{1791}:C_{66}$ ($C_{66}$), $C_{1194}:C_{99}$ ($C_{66}$), $C_{597}:C_{198}$ ($C_{66}$), $C_3\times F_{199}$ ($C_{66}$), $C_{1194}:C_{99}$ ($C_{66}$)

Order 96316: $C_{4378}:C_{22}$ ($C_9^2$)

Order 78804: $C_{1194}:C_{66}$ ($C_3\times C_{33}$), $C_{1194}:C_{66}$ ($C_3\times C_{33}$), $C_{398}:C_{198}$ ($C_{99}$), $C_{3582}:C_{22}$ ($C_{99}$), $C_{398}:C_{198}$ ($C_{99}$), $C_{99}\times D_{398}$ ($C_{99}$), $C_{398}:C_{198}$ ($C_{99}$), $C_2\times F_{199}$ ($C_{99}$)

Order 72237: $C_{2189}:C_{33}$ ($C_6\times C_{18}$), $C_{2189}:C_{33}$ ($C_6\times C_{18}$)

Order 64476: $C_{3582}:C_{18}$ ($C_{11}^2$)

Order 59103: $C_{597}:C_{99}$ ($C_2\times C_{66}$), $C_{1791}:C_{33}$ ($C_2\times C_{66}$), $C_{1791}:C_{33}$ ($C_2\times C_{66}$), $C_{597}:C_{99}$ ($C_2\times C_{66}$)

Order 48158: $C_{2189}:C_{22}$ ($C_9\times C_{18}$), $C_{2189}:C_{22}$ ($C_9\times C_{18}$)

Order 39402: $C_{66}\times C_{199}:C_3$ ($C_3\times C_{66}$), $C_{99}\times D_{199}$ ($C_{198}$), $C_{39402}$ ($C_{198}$), $C_{199}:C_{198}$ ($C_{198}$), $C_{199}:C_{198}$ ($C_{198}$), $C_{597}:C_{66}$ ($C_3\times C_{66}$), $C_{199}:C_{198}$ ($C_{198}$), $C_{1791}:C_{22}$ ($C_{198}$), $C_{1791}:C_{22}$ ($C_{198}$), $C_{199}:C_{198}$ ($C_{198}$), $C_{199}:C_{198}$ ($C_{198}$), $C_{199}:C_{198}$ ($C_{198}$), $F_{199}$ ($C_{198}$), $C_{1194}:C_{33}$ ($C_3\times C_{66}$), $C_{597}:C_{66}$ ($C_3\times C_{66}$), $C_{199}:C_{198}$ ($C_{198}$)

Order 32238: $C_{1791}:C_{18}$ ($C_{11}\times C_{22}$), $C_{1791}:C_{18}$ ($C_{11}\times C_{22}$)

Order 26268: $C_{66}\times D_{199}$ ($C_3\times C_{99}$), $C_{398}:C_{66}$ ($C_3\times C_{99}$), $C_{1194}:C_{22}$ ($C_3\times C_{99}$), $C_{398}:C_{66}$ ($C_3\times C_{99}$)

Order 24079: $C_{2189}:C_{11}$ ($C_{18}^2$)

Order 21492: $C_{3582}:C_6$ ($C_{11}\times C_{33}$), $C_{1194}:C_{18}$ ($C_{11}\times C_{33}$)

Order 19701: $C_{597}:C_{33}$ ($C_6\times C_{66}$), $C_{199}:C_{99}$ ($C_2\times C_{198}$), $C_{199}:C_{99}$ ($C_2\times C_{198}$), $C_{597}:C_{33}$ ($C_6\times C_{66}$), $C_{1791}:C_{11}$ ($C_2\times C_{198}$), $C_{199}:C_{99}$ ($C_2\times C_{198}$), $C_{199}:C_{99}$ ($C_2\times C_{198}$), $C_{19701}$ ($C_2\times C_{198}$)

Order 16119: $C_{1791}:C_9$ ($C_{22}^2$)

Order 13134: $C_{199}:C_{66}$ ($C_3\times C_{198}$), $C_{199}:C_{66}$ ($C_3\times C_{198}$), $C_{199}:C_{66}$ ($C_3\times C_{198}$), $C_{33}\times D_{199}$ ($C_3\times C_{198}$), $C_{199}:C_{66}$ ($C_3\times C_{198}$), $C_{199}:C_{66}$ ($C_3\times C_{198}$), $C_{1194}:C_{11}$ ($C_3\times C_{198}$), $C_{13134}$ ($C_3\times C_{198}$)

Order 10746: $C_{1791}:C_6$ ($C_{11}\times C_{66}$), $C_{1791}:C_6$ ($C_{11}\times C_{66}$), $C_{1194}:C_9$ ($C_{11}\times C_{66}$), $C_{597}:C_{18}$ ($C_{11}\times C_{66}$)

Order 8756: $C_{11}\times D_{398}$ ($C_9\times C_{99}$), $C_{398}:C_{22}$ ($C_9\times C_{99}$)

Order 7164: $C_{1194}:C_6$ ($C_{33}^2$), $C_9\times D_{398}$ ($C_{11}\times C_{99}$), $C_{398}:C_{18}$ ($C_{11}\times C_{99}$), $C_{398}:C_{18}$ ($C_{11}\times C_{99}$)

Order 6567: $C_{6567}$ ($C_6\times C_{198}$), $C_{199}:C_{33}$ ($C_6\times C_{198}$), $C_{199}:C_{33}$ ($C_6\times C_{198}$), $C_{199}:C_{33}$ ($C_6\times C_{198}$)

Order 5373: $C_{1791}:C_3$ ($C_{22}\times C_{66}$), $C_{597}:C_9$ ($C_{22}\times C_{66}$)

Order 4378: $C_{4378}$ ($C_9\times C_{198}$), $C_{11}\times D_{199}$ ($C_9\times C_{198}$), $C_{199}:C_{22}$ ($C_9\times C_{198}$), $C_{199}:C_{22}$ ($C_9\times C_{198}$)

Order 3582: $C_{1194}:C_3$ ($C_{33}\times C_{66}$), $C_{199}:C_{18}$ ($C_{11}\times C_{198}$), $C_{199}:C_{18}$ ($C_{11}\times C_{198}$), $C_{597}:C_6$ ($C_{33}\times C_{66}$), $C_9\times D_{199}$ ($C_{11}\times C_{198}$), $C_{199}:C_{18}$ ($C_{11}\times C_{198}$), $C_{199}:C_{18}$ ($C_{11}\times C_{198}$), $C_{3582}$ ($C_{11}\times C_{198}$)

Order 2388: $C_3\times D_{398}$ ($C_{33}\times C_{99}$), $C_{398}:C_6$ ($C_{33}\times C_{99}$)

Order 2189: $C_{2189}$ ($C_{18}\times C_{198}$), $C_{199}:C_{11}$ ($C_{18}\times C_{198}$)

Order 1791: $C_{597}:C_3$ ($C_{66}^2$), $C_{199}:C_9$ ($C_{22}\times C_{198}$), $C_{1791}$ ($C_{22}\times C_{198}$), $C_{199}:C_9$ ($C_{22}\times C_{198}$)

Order 1194: $C_{1194}$ ($C_{33}\times C_{198}$), $C_3\times D_{199}$ ($C_{33}\times C_{198}$), $C_{199}:C_6$ ($C_{33}\times C_{198}$), $C_{199}:C_6$ ($C_{33}\times C_{198}$)

Order 796: $D_{398}$ ($C_{99}^2$)

Order 597: $C_{597}$ ($C_{66}\times C_{198}$), $C_{199}:C_3$ ($C_{66}\times C_{198}$)

Order 398: $C_{398}$ ($C_{99}\times C_{198}$), $D_{199}$ ($C_{99}\times C_{198}$)

Order 199: $C_{199}$ ($C_{198}^2$)

Order 198: $C_{198}$ ($F_{199}$)

Order 99: $C_{99}$ ($C_2\times F_{199}$)

Order 66: $C_{66}$ ($C_3\times F_{199}$)

Order 33: $C_{33}$ ($C_6\times F_{199}$)

Order 22: $C_{22}$ ($C_9\times F_{199}$)

Order 18: $C_{18}$ ($C_{11}\times F_{199}$)

Order 11: $C_{11}$ ($C_{18}\times F_{199}$)

Order 9: $C_9$ ($C_{22}\times F_{199}$)

Order 6: $C_6$ ($C_{33}\times F_{199}$)

Order 3: $C_3$ ($C_{66}\times F_{199}$)

Order 2: $C_2$ ($C_{99}\times F_{199}$)

Order 1: $C_1$ ($C_{199}:C_{198}^2$)

Series

Derived series

$C_{199}:C_{198}^2$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Chief series

$C_{199}:C_{198}^2$

$\rhd$

$C_{19701}:C_{198}$

$\rhd$

$C_{19701}:C_{66}$

$\rhd$

$C_{3582}:C_{11}^2$

$\rhd$

$C_{39402}$

$\rhd$

$C_{19701}$

$\rhd$

$C_{6567}$

$\rhd$

$C_{2189}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Lower central series

$C_{199}:C_{198}^2$

$\rhd$

$C_{199}$

Upper central series

$C_1$

$\lhd$

$C_{198}$

Character theory

Complex character table

The $39402 \times 39402$ character table is not available for this group.

Rational character table

The $896 \times 896$ rational character table is not available for this group.