Abstract group 118206.b: $C_3\times F_{199}$

• Label: 118206.b
• Order: \( 2 \cdot 3^{3} \cdot 11 \cdot 199 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{3} \cdot 11 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{3} \cdot 11 \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $202$
• Trans deg.: $597$
• Rank: $2$

Group information

Description:	$C_3\times F_{199}$
Order:	\(118206\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \cdot 199 \)
Exponent:	\(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \)
Automorphism group:	$C_{199}:(C_{11}:(C_{18}\times S_3))$, of order \(236412\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 11 \cdot 199 \)
Composition factors:	$C_2$, $C_3$ x 3, $C_{11}$, $C_{199}$
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
Elements	1	199	1196	1592	3582	1990	3582	1990	15920	15920	35820	35820	198	396	118206
Conjugacy classes	1	1	8	8	18	10	18	10	80	80	180	180	1	2	597
Divisions	1	1	4	4	3	1	3	1	4	4	3	3	1	1	34
Autjugacy classes	1	1	5	5	6	10	6	10	50	50	60	60	1	1	266

Dimension	1	2	6	10	20	60	198	396
Irr. complex chars.	594	0	0	0	0	0	3	0	597
Irr. rational chars.	2	8	6	2	8	6	1	1	34

Minimal presentations

Permutation degree:	$202$
Transitive degree:	$597$
Rank:	$2$
Inequivalent generating pairs:	$25920$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	198	not computed	396
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{198}=b^{597}=1, b^{a}=b^{436} \rangle$
Permutation group:	Degree $202$ $\langle(2,4,22,100,199,178,192,89,17,3,13,72,190,158,148,124,59,9)(5,23,53,169,193,165,171,128,47,14,69,87,175,114,196,182,144,35) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 42 & 0 \\ 0 & 128 \end{array}\right), \left(\begin{array}{rr} 106 & 0 \\ 0 & 92 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_3$ $\, \times\, $ $F_{199}$
Semidirect product:	$C_{597}$ $\,\rtimes\,$ $C_{198}$	$(C_3\times D_{199})$ $\,\rtimes\,$ $C_{99}$	$D_{199}$ $\,\rtimes\,$ $(C_3\times C_{99})$	$(C_{597}:C_{99})$ $\,\rtimes\,$ $C_2$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{597}:C_{66})$ . $C_3$	$(C_{597}:C_{33})$ . $C_6$	$(C_{597}:C_6)$ . $C_{33}$	$(C_{597}:C_3)$ . $C_{66}$	all 8
Aut. group:	$\Aut(C_{199}:C_9)$	$\Aut(C_{199}:C_{18})$	$\Aut(C_{199}:C_{18})$	$\Aut(C_{199}:C_{99})$	all 6

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

Homology

Abelianization:	$C_{3} \times C_{198} \simeq C_{2} \times C_{3} \times C_{9} \times C_{11}$
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 7604 subgroups in 80 conjugacy classes, 42 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $F_{199}$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_3\times C_{198}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3\times F_{199}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{597}$	$G/\operatorname{Fit} \simeq$ $C_{198}$
Radical:	$R \simeq$ $C_3\times F_{199}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{597}$	$G/\operatorname{soc} \simeq$ $C_{198}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
199-Sylow subgroup:	$P_{ 199 } \simeq$ $C_{199}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 118206: $C_3\times F_{199}$

Order 59103: $C_{597}:C_{99}$

Order 39402: $F_{199}$ x 3, $C_{597}:C_{66}$

Order 19701: $C_{199}:C_{99}$ x 3, $C_{597}:C_{33}$

Order 13134: $C_{199}:C_{66}$ x 3, $C_{199}:C_{66}$

Order 10746: $C_{597}:C_{18}$

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

Order 5373: $C_{597}:C_9$

Order 4378: $C_{199}:C_{22}$

Order 3582: $C_{199}:C_{18}$ x 3, $C_{597}:C_6$

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

Order 1791: $C_{199}:C_9$ x 3, $C_{597}:C_3$

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

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

Order 594: $C_3\times C_{198}$

Order 398: $D_{199}$

Order 297: $C_3\times C_{99}$

Order 199: $C_{199}$

Order 198: $C_{198}$ x 3, $C_3\times C_{66}$

Order 99: $C_{99}$ x 3, $C_3\times C_{33}$

Order 66: $C_{66}$ x 4

Order 54: $C_3\times C_{18}$

Order 33: $C_{33}$ x 4

Order 27: $C_3\times C_9$

Order 22: $C_{22}$

Order 18: $C_{18}$ x 3, $C_3\times C_6$

Order 11: $C_{11}$

Order 9: $C_9$ x 3, $C_3^2$

Order 6: $C_6$ x 4

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 118206: $C_3\times F_{199}$

Order 59103: $C_{597}:C_{99}$

Order 39402: $F_{199}$, $C_{597}:C_{66}$

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

Order 13134: $C_{199}:C_{66}$ x 2, $C_{199}:C_{66}$

Order 10746: $C_{597}:C_{18}$

Order 6567: $C_{199}:C_{33}$ x 2, $C_{199}:C_{33}$

Order 5373: $C_{597}:C_9$

Order 4378: $C_{199}:C_{22}$

Order 3582: $C_{597}:C_6$, $C_{199}:C_{18}$

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

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

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

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

Order 594: $C_3\times C_{198}$

Order 398: $D_{199}$

Order 297: $C_3\times C_{99}$

Order 199: $C_{199}$

Order 198: $C_{198}$, $C_3\times C_{66}$

Order 99: $C_3\times C_{33}$, $C_{99}$

Order 66: $C_{66}$ x 3

Order 54: $C_3\times C_{18}$

Order 33: $C_{33}$ x 3

Order 27: $C_3\times C_9$

Order 22: $C_{22}$

Order 18: $C_3\times C_6$, $C_{18}$

Order 11: $C_{11}$

Order 9: $C_3^2$, $C_9$

Order 6: $C_6$ x 3

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 118206: $C_3\times F_{199}$ ($C_1$)

Order 59103: $C_{597}:C_{99}$ ($C_2$)

Order 39402: $F_{199}$ ($C_3$) x 3, $C_{597}:C_{66}$ ($C_3$)

Order 19701: $C_{199}:C_{99}$ ($C_6$) x 3, $C_{597}:C_{33}$ ($C_6$)

Order 13134: $C_{199}:C_{66}$ ($C_9$) x 2, $C_{199}:C_{66}$ ($C_3^2$), $C_{199}:C_{66}$ ($C_9$)

Order 10746: $C_{597}:C_{18}$ ($C_{11}$)

Order 6567: $C_{199}:C_{33}$ ($C_{18}$) x 2, $C_{199}:C_{33}$ ($C_{18}$), $C_{199}:C_{33}$ ($C_3\times C_6$)

Order 5373: $C_{597}:C_9$ ($C_{22}$)

Order 4378: $C_{199}:C_{22}$ ($C_3\times C_9$)

Order 3582: $C_{199}:C_{18}$ ($C_{33}$) x 3, $C_{597}:C_6$ ($C_{33}$)

Order 2189: $C_{199}:C_{11}$ ($C_3\times C_{18}$)

Order 1791: $C_{199}:C_9$ ($C_{66}$) x 3, $C_{597}:C_3$ ($C_{66}$)

Order 1194: $C_{199}:C_6$ ($C_{99}$) x 2, $C_3\times D_{199}$ ($C_{99}$), $C_{199}:C_6$ ($C_3\times C_{33}$)

Order 597: $C_{199}:C_3$ ($C_{198}$) x 2, $C_{597}$ ($C_{198}$), $C_{199}:C_3$ ($C_3\times C_{66}$)

Order 398: $D_{199}$ ($C_3\times C_{99}$)

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

Order 3: $C_3$ ($F_{199}$)

Order 1: $C_1$ ($C_3\times F_{199}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 118206: $C_3\times F_{199}$ ($C_1$)

Order 59103: $C_{597}:C_{99}$ ($C_2$)

Order 39402: $F_{199}$ ($C_3$), $C_{597}:C_{66}$ ($C_3$)

Order 19701: $C_{597}:C_{33}$ ($C_6$), $C_{199}:C_{99}$ ($C_6$)

Order 13134: $C_{199}:C_{66}$ ($C_3^2$), $C_{199}:C_{66}$ ($C_9$), $C_{199}:C_{66}$ ($C_9$)

Order 10746: $C_{597}:C_{18}$ ($C_{11}$)

Order 6567: $C_{199}:C_{33}$ ($C_{18}$), $C_{199}:C_{33}$ ($C_3\times C_6$), $C_{199}:C_{33}$ ($C_{18}$)

Order 5373: $C_{597}:C_9$ ($C_{22}$)

Order 4378: $C_{199}:C_{22}$ ($C_3\times C_9$)

Order 3582: $C_{597}:C_6$ ($C_{33}$), $C_{199}:C_{18}$ ($C_{33}$)

Order 2189: $C_{199}:C_{11}$ ($C_3\times C_{18}$)

Order 1791: $C_{597}:C_3$ ($C_{66}$), $C_{199}:C_9$ ($C_{66}$)

Order 1194: $C_3\times D_{199}$ ($C_{99}$), $C_{199}:C_6$ ($C_3\times C_{33}$), $C_{199}:C_6$ ($C_{99}$)

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

Order 398: $D_{199}$ ($C_3\times C_{99}$)

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

Order 3: $C_3$ ($F_{199}$)

Order 1: $C_1$ ($C_3\times F_{199}$)

Series

Derived series

$C_3\times F_{199}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Chief series

$C_3\times F_{199}$

$\rhd$

$C_{597}:C_{99}$

$\rhd$

$C_{597}:C_{33}$

$\rhd$

$C_{199}:C_{33}$

$\rhd$

$C_{597}$

$\rhd$

$C_{199}$

$\rhd$

$C_1$

Lower central series

$C_3\times F_{199}$

$\rhd$

$C_{199}$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

The $597 \times 597$ character table is not available for this group.

Rational character table

See the $34 \times 34$ rational character table.