Abstract group 433422.e: $C_{2189}:C_{198}$

• Label: 433422.e
• Order: \( 2 \cdot 3^{2} \cdot 11^{2} \cdot 199 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \cdot 11^{2} \)
• $\card{Z(G)}$: \( 3 \cdot 11 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 11^{2} \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
• Perm deg.: $219$
• Trans deg.: $19701$
• Rank: $2$

Group information

Description:	$C_{2189}:C_{198}$
Order:	\(433422\)\(\medspace = 2 \cdot 3^{2} \cdot 11^{2} \cdot 199 \)
Exponent:	\(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Automorphism group:	$C_{2189}.C_{165}.C_6^2$, of order \(13002660\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \cdot 11^{2} \cdot 199 \)
Composition factors:	$C_2$, $C_3$ x 2, $C_{11}$ 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	597	2189	6567
Elements	1	199	2	398	1194	21900	1194	23880	43800	47760	143280	143280	198	396	1980	3960	433422
Conjugacy classes	1	1	2	2	6	120	6	120	240	240	720	720	3	6	30	60	2277
Divisions	1	1	1	1	1	12	1	12	12	12	12	12	1	1	1	1	82
Autjugacy classes	1	1	2	2	2	11	2	11	22	22	22	22	1	2	1	2	126

Minimal presentations

Permutation degree:	$219$
Transitive degree:	$19701$
Rank:	$2$
Inequivalent generating pairs:	$8640$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	66	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{198}=b^{2189}=1, b^{a}=b^{1651} \rangle$
Permutation group:	Degree $219$ $\langle(2,4,14,56,43,124,156,101,102,29,7)(3,9,35,89,24,87,153,128,114,45,11)(5,19,71,77,121,196,192,155,176,82,22) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 133 \end{array}\right), \left(\begin{array}{rr} 125 & 0 \\ 0 & 121 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_{11}$ $\, \times\, $ $(C_{199}:C_{198})$
Semidirect product:	$C_{2189}$ $\,\rtimes\,$ $C_{198}$	$(C_{11}\times D_{199})$ $\,\rtimes\,$ $C_{99}$	$D_{199}$ $\,\rtimes\,$ $(C_{11}\times C_{99})$	$(C_{2189}:C_{22})$ $\,\rtimes\,$ $C_9$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{6567}$ . $C_{66}$	$(C_{33}\times D_{199})$ . $C_{33}$	$(C_{2189}:C_{33})$ . $C_6$	$(C_{199}:C_{66})$ . $C_{33}$	all 10

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

Homology

Abelianization:	$C_{11} \times C_{198} \simeq C_{2} \times C_{9} \times C_{11}^{2}$
Schur multiplier:	$C_{11}$
Commutator length:	$1$

Subgroups

There are 16008 subgroups in 168 conjugacy classes, 88 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{33}$	$G/Z \simeq$ $C_{199}:C_{66}$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_{11}\times C_{198}$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_{199}:(C_{11}\times C_{66})$
Fitting:	$\operatorname{Fit} \simeq$ $C_{6567}$	$G/\operatorname{Fit} \simeq$ $C_{66}$
Radical:	$R \simeq$ $C_{2189}:C_{198}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{6567}$	$G/\operatorname{soc} \simeq$ $C_{66}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
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: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 433422: $C_{2189}:C_{198}$

Order 216711: $C_{11}\times C_{199}:C_{99}$

Order 144474: $C_{33}\times C_{199}:C_{22}$

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

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

Order 39402: $C_{199}:C_{198}$ x 11, $C_{11}\times C_{199}:C_{18}$

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

Order 19701: $C_{199}:C_{99}$ x 11, $C_{199}:C_{99}$

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

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

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

Order 3582: $C_{199}:C_{18}$

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

Order 2178: $C_{11}\times C_{198}$

Order 1791: $C_{199}:C_9$

Order 1194: $C_3\times D_{199}$

Order 1089: $C_{11}\times C_{99}$

Order 726: $C_{11}\times C_{66}$

Order 597: $C_{597}$

Order 398: $D_{199}$

Order 363: $C_{11}\times C_{33}$

Order 242: $C_{11}\times C_{22}$

Order 199: $C_{199}$

Order 198: $C_{198}$ x 12

Order 121: $C_{11}^2$

Order 99: $C_{99}$ x 12

Order 66: $C_{66}$ x 12

Order 33: $C_{33}$ x 12

Order 22: $C_{22}$ x 12

Order 18: $C_{18}$

Order 11: $C_{11}$ x 12

Order 9: $C_9$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 433422: $C_{2189}:C_{198}$

Order 216711: $C_{11}\times C_{199}:C_{99}$

Order 144474: $C_{33}\times C_{199}:C_{22}$

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

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

Order 39402: $C_{199}:C_{198}$, $C_{11}\times C_{199}:C_{18}$

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

Order 19701: $C_{199}:C_{99}$, $C_{199}:C_{99}$

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

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

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

Order 3582: $C_{199}:C_{18}$

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

Order 2178: $C_{11}\times C_{198}$

Order 1791: $C_{199}:C_9$

Order 1194: $C_3\times D_{199}$

Order 1089: $C_{11}\times C_{99}$

Order 726: $C_{11}\times C_{66}$

Order 597: $C_{597}$

Order 398: $D_{199}$

Order 363: $C_{11}\times C_{33}$

Order 242: $C_{11}\times C_{22}$

Order 199: $C_{199}$

Order 198: $C_{198}$ x 2

Order 121: $C_{11}^2$

Order 99: $C_{99}$ x 2

Order 66: $C_{66}$ x 2

Order 33: $C_{33}$ x 2

Order 22: $C_{22}$ x 2

Order 18: $C_{18}$

Order 11: $C_{11}$ x 2

Order 9: $C_9$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 433422: $C_{2189}:C_{198}$ ($C_1$)

Order 216711: $C_{11}\times C_{199}:C_{99}$ ($C_2$)

Order 144474: $C_{33}\times C_{199}:C_{22}$ ($C_3$)

Order 72237: $C_{2189}:C_{33}$ ($C_6$)

Order 48158: $C_{2189}:C_{22}$ ($C_9$)

Order 39402: $C_{199}:C_{198}$ ($C_{11}$) x 11, $C_{11}\times C_{199}:C_{18}$ ($C_{11}$)

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

Order 19701: $C_{199}:C_{99}$ ($C_{22}$) x 11, $C_{199}:C_{99}$ ($C_{22}$)

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

Order 6567: $C_{199}:C_{33}$ ($C_{66}$) x 11, $C_{6567}$ ($C_{66}$)

Order 4378: $C_{199}:C_{22}$ ($C_{99}$) x 11, $C_{11}\times D_{199}$ ($C_{99}$)

Order 3582: $C_{199}:C_{18}$ ($C_{11}^2$)

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

Order 1791: $C_{199}:C_9$ ($C_{11}\times C_{22}$)

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

Order 597: $C_{597}$ ($C_{11}\times C_{66}$)

Order 398: $D_{199}$ ($C_{11}\times C_{99}$)

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

Order 33: $C_{33}$ ($C_{199}:C_{66}$)

Order 11: $C_{11}$ ($C_{199}:C_{198}$)

Order 3: $C_3$ ($C_{199}:(C_{11}\times C_{66})$)

Order 1: $C_1$ ($C_{2189}:C_{198}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 433422: $C_{2189}:C_{198}$ ($C_1$)

Order 216711: $C_{11}\times C_{199}:C_{99}$ ($C_2$)

Order 144474: $C_{33}\times C_{199}:C_{22}$ ($C_3$)

Order 72237: $C_{2189}:C_{33}$ ($C_6$)

Order 48158: $C_{2189}:C_{22}$ ($C_9$)

Order 39402: $C_{199}:C_{198}$ ($C_{11}$), $C_{11}\times C_{199}:C_{18}$ ($C_{11}$)

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

Order 19701: $C_{199}:C_{99}$ ($C_{22}$), $C_{199}:C_{99}$ ($C_{22}$)

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

Order 6567: $C_{6567}$ ($C_{66}$), $C_{199}:C_{33}$ ($C_{66}$)

Order 4378: $C_{11}\times D_{199}$ ($C_{99}$), $C_{199}:C_{22}$ ($C_{99}$)

Order 3582: $C_{199}:C_{18}$ ($C_{11}^2$)

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

Order 1791: $C_{199}:C_9$ ($C_{11}\times C_{22}$)

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

Order 597: $C_{597}$ ($C_{11}\times C_{66}$)

Order 398: $D_{199}$ ($C_{11}\times C_{99}$)

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

Order 33: $C_{33}$ ($C_{199}:C_{66}$)

Order 11: $C_{11}$ ($C_{199}:C_{198}$)

Order 3: $C_3$ ($C_{199}:(C_{11}\times C_{66})$)

Order 1: $C_1$ ($C_{2189}:C_{198}$)

Series

Derived series

$C_{2189}:C_{198}$

$\rhd$

$C_{2189}:C_{198}$

$\rhd$

$C_{199}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{2189}:C_{198}$

$\rhd$

$C_{2189}:C_{198}$

$\rhd$

$C_{11}\times C_{199}:C_{99}$

$\rhd$

$C_{11}\times C_{199}:C_{99}$

$\rhd$

$C_{2189}:C_{33}$

$\rhd$

$C_{2189}:C_{33}$

$\rhd$

$C_{2189}:C_{11}$

$\rhd$

$C_{2189}:C_{11}$

$\rhd$

$C_{2189}$

$\rhd$

$C_{2189}$

$\rhd$

$C_{199}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{2189}:C_{198}$

$\rhd$

$C_{2189}:C_{198}$

$\rhd$

$C_{199}$

$\rhd$

$C_{199}$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_{33}$

$\lhd$

$C_{33}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $2277 \times 2277$ character table is not available for this group.

Rational character table

The $82 \times 82$ rational character table is not available for this group.