Abstract group 13134.f: $C_{199}:C_{66}$

• Label: 13134.f
• Order: \( 2 \cdot 3 \cdot 11 \cdot 199 \)
• Exponent: \( 2 \cdot 3 \cdot 11 \cdot 199 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \cdot 11 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{2} \cdot 11 \cdot 199 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \)
• Perm deg.: $202$
• Trans deg.: $597$
• Rank: $2$

Group information

Description:	$C_{199}:C_{66}$
Order:	\(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \)
Exponent:	\(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \)
Automorphism group:	$C_2\times F_{199}$, of order \(78804\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \cdot 199 \)
Composition factors:	$C_2$, $C_3$, $C_{11}$, $C_{199}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	11	22	33	66	199	597
Elements	1	199	2	398	1990	1990	3980	3980	198	396	13134
Conjugacy classes	1	1	2	2	10	10	20	20	9	18	93
Divisions	1	1	1	1	1	1	1	1	1	1	10
Autjugacy classes	1	1	1	1	10	10	10	10	1	1	46

Minimal presentations

Permutation degree:	$202$
Transitive degree:	$597$
Rank:	$2$
Inequivalent generating pairs:	$1440$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	22	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{22}=b^{597}=1, b^{a}=b^{472} \rangle$
Permutation group:	Degree $202$ $\langle(2,3)(4,7)(5,9)(6,8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,24)(17,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 59 & 0 \\ 0 & 132 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_3$ $\, \times\, $ $(C_{199}:C_{22})$
Semidirect product:	$D_{199}$ $\,\rtimes\,$ $C_{33}$	$C_{597}$ $\,\rtimes\,$ $C_{22}$	$C_{199}$ $\,\rtimes\,$ $C_{66}$	$(C_3\times D_{199})$ $\,\rtimes\,$ $C_{11}$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{66} \simeq C_{2} \times C_{3} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 1204 subgroups in 16 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_{199}:C_{22}$
Commutator:	$G' \simeq$ $C_{199}$	$G/G' \simeq$ $C_{66}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{199}:C_{66}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{597}$	$G/\operatorname{Fit} \simeq$ $C_{22}$
Radical:	$R \simeq$ $C_{199}:C_{66}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{597}$	$G/\operatorname{soc} \simeq$ $C_{22}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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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Classes of subgroups up to conjugation

Order 13134: $C_{199}:C_{66}$

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

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

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

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

Order 597: $C_{597}$

Order 398: $D_{199}$

Order 199: $C_{199}$

Order 66: $C_{66}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 13134: $C_{199}:C_{66}$

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

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

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

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

Order 597: $C_{597}$

Order 398: $D_{199}$

Order 199: $C_{199}$

Order 66: $C_{66}$

Order 33: $C_{33}$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 13134: $C_{199}:C_{66}$ ($C_1$)

Order 6567: $C_{199}:C_{33}$ ($C_2$)

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

Order 2189: $C_{199}:C_{11}$ ($C_6$)

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

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

Order 398: $D_{199}$ ($C_{33}$)

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

Order 3: $C_3$ ($C_{199}:C_{22}$)

Order 1: $C_1$ ($C_{199}:C_{66}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 13134: $C_{199}:C_{66}$ ($C_1$)

Order 6567: $C_{199}:C_{33}$ ($C_2$)

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

Order 2189: $C_{199}:C_{11}$ ($C_6$)

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

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

Order 398: $D_{199}$ ($C_{33}$)

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

Order 3: $C_3$ ($C_{199}:C_{22}$)

Order 1: $C_1$ ($C_{199}:C_{66}$)

Series

Derived series	$C_{199}:C_{66}$	$\rhd$	$C_{199}$	$\rhd$	$C_1$
Chief series	$C_{199}:C_{66}$	$\rhd$	$C_{199}:C_{33}$	$\rhd$	$C_{597}$	$\rhd$	$C_{199}$	$\rhd$	$C_1$
Lower central series	$C_{199}:C_{66}$	$\rhd$	$C_{199}$
Upper central series	$C_1$	$\lhd$	$C_3$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

The $93 \times 93$ character table is not available for this group.

Rational character table

The $10 \times 10$ rational character table is not available for this group.