Abstract group 11610.b: $C_{11610}$

• Label: 11610.b
• Order: \( 2 \cdot 3^{3} \cdot 5 \cdot 43 \)
• Exponent: \( 2 \cdot 3^{3} \cdot 5 \cdot 43 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 7 \)
• Perm deg.: $77$
• Trans deg.: $11610$
• Rank: $1$

Group information

Description:	$C_{11610}$
Order:	\(11610\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \cdot 43 \)
Exponent:	\(11610\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \cdot 43 \)
Automorphism group:	$C_2\times C_6\times C_{252}$, of order \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$, $C_3$ x 3, $C_5$, $C_{43}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	3	5	6	9	10	15	18	27	30	43	45	54	86	90	129	135	215	258	270	387	430	645	774	1161	1290	1935	2322	3870	5805	11610
Elements	1	1	2	4	2	6	4	8	6	18	8	42	24	18	42	24	84	72	168	84	72	252	168	336	252	756	336	1008	756	1008	3024	3024	11610
Conjugacy classes	1	1	2	4	2	6	4	8	6	18	8	42	24	18	42	24	84	72	168	84	72	252	168	336	252	756	336	1008	756	1008	3024	3024	11610
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	32
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	32

Minimal presentations

Permutation degree:	$77$
Transitive degree:	$11610$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	not computed	not computed
Arbitrary	1	not computed	not computed

Constructions

Presentation:	$\langle a \mid a^{11610}=1 \rangle$
Permutation group:	Degree $77$ $\langle(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 200 & 368 \\ 422 & 200 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{431})$
Direct product:	$C_2$ $\, \times\, $ $C_{27}$ $\, \times\, $ $C_5$ $\, \times\, $ $C_{43}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{3870}$ . $C_3$	$C_{1935}$ . $C_6$	$C_{1290}$ . $C_9$	$C_{774}$ . $C_{15}$	all 16

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

Homology

Primary decomposition:	$C_{2} \times C_{27} \times C_{5} \times C_{43}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 32 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{11610}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{11610}$
Frattini:	$\Phi \simeq$ $C_9$	$G/\Phi \simeq$ $C_{1290}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11610}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{11610}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1290}$	$G/\operatorname{soc} \simeq$ $C_9$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_{27}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
43-Sylow subgroup:	$P_{ 43 } \simeq$ $C_{43}$

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 11610: $C_{11610}$

Order 5805: $C_{5805}$

Order 3870: $C_{3870}$

Order 2322: $C_{2322}$

Order 1935: $C_{1935}$

Order 1290: $C_{1290}$

Order 1161: $C_{1161}$

Order 774: $C_{774}$

Order 645: $C_{645}$

Order 430: $C_{430}$

Order 387: $C_{387}$

Order 270: $C_{270}$

Order 258: $C_{258}$

Order 215: $C_{215}$

Order 135: $C_{135}$

Order 129: $C_{129}$

Order 90: $C_{90}$

Order 86: $C_{86}$

Order 54: $C_{54}$

Order 45: $C_{45}$

Order 43: $C_{43}$

Order 30: $C_{30}$

Order 27: $C_{27}$

Order 18: $C_{18}$

Order 15: $C_{15}$

Order 10: $C_{10}$

Order 9: $C_9$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11610: $C_{11610}$

Order 5805: $C_{5805}$

Order 3870: $C_{3870}$

Order 2322: $C_{2322}$

Order 1935: $C_{1935}$

Order 1290: $C_{1290}$

Order 1161: $C_{1161}$

Order 774: $C_{774}$

Order 645: $C_{645}$

Order 430: $C_{430}$

Order 387: $C_{387}$

Order 270: $C_{270}$

Order 258: $C_{258}$

Order 215: $C_{215}$

Order 135: $C_{135}$

Order 129: $C_{129}$

Order 90: $C_{90}$

Order 86: $C_{86}$

Order 54: $C_{54}$

Order 45: $C_{45}$

Order 43: $C_{43}$

Order 30: $C_{30}$

Order 27: $C_{27}$

Order 18: $C_{18}$

Order 15: $C_{15}$

Order 10: $C_{10}$

Order 9: $C_9$

Order 6: $C_6$

Order 5: $C_5$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11610: $C_{11610}$ ($C_1$)

Order 5805: $C_{5805}$ ($C_2$)

Order 3870: $C_{3870}$ ($C_3$)

Order 2322: $C_{2322}$ ($C_5$)

Order 1935: $C_{1935}$ ($C_6$)

Order 1290: $C_{1290}$ ($C_9$)

Order 1161: $C_{1161}$ ($C_{10}$)

Order 774: $C_{774}$ ($C_{15}$)

Order 645: $C_{645}$ ($C_{18}$)

Order 430: $C_{430}$ ($C_{27}$)

Order 387: $C_{387}$ ($C_{30}$)

Order 270: $C_{270}$ ($C_{43}$)

Order 258: $C_{258}$ ($C_{45}$)

Order 215: $C_{215}$ ($C_{54}$)

Order 135: $C_{135}$ ($C_{86}$)

Order 129: $C_{129}$ ($C_{90}$)

Order 90: $C_{90}$ ($C_{129}$)

Order 86: $C_{86}$ ($C_{135}$)

Order 54: $C_{54}$ ($C_{215}$)

Order 45: $C_{45}$ ($C_{258}$)

Order 43: $C_{43}$ ($C_{270}$)

Order 30: $C_{30}$ ($C_{387}$)

Order 27: $C_{27}$ ($C_{430}$)

Order 18: $C_{18}$ ($C_{645}$)

Order 15: $C_{15}$ ($C_{774}$)

Order 10: $C_{10}$ ($C_{1161}$)

Order 9: $C_9$ ($C_{1290}$)

Order 6: $C_6$ ($C_{1935}$)

Order 5: $C_5$ ($C_{2322}$)

Order 3: $C_3$ ($C_{3870}$)

Order 2: $C_2$ ($C_{5805}$)

Order 1: $C_1$ ($C_{11610}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11610: $C_{11610}$ ($C_1$)

Order 5805: $C_{5805}$ ($C_2$)

Order 3870: $C_{3870}$ ($C_3$)

Order 2322: $C_{2322}$ ($C_5$)

Order 1935: $C_{1935}$ ($C_6$)

Order 1290: $C_{1290}$ ($C_9$)

Order 1161: $C_{1161}$ ($C_{10}$)

Order 774: $C_{774}$ ($C_{15}$)

Order 645: $C_{645}$ ($C_{18}$)

Order 430: $C_{430}$ ($C_{27}$)

Order 387: $C_{387}$ ($C_{30}$)

Order 270: $C_{270}$ ($C_{43}$)

Order 258: $C_{258}$ ($C_{45}$)

Order 215: $C_{215}$ ($C_{54}$)

Order 135: $C_{135}$ ($C_{86}$)

Order 129: $C_{129}$ ($C_{90}$)

Order 90: $C_{90}$ ($C_{129}$)

Order 86: $C_{86}$ ($C_{135}$)

Order 54: $C_{54}$ ($C_{215}$)

Order 45: $C_{45}$ ($C_{258}$)

Order 43: $C_{43}$ ($C_{270}$)

Order 30: $C_{30}$ ($C_{387}$)

Order 27: $C_{27}$ ($C_{430}$)

Order 18: $C_{18}$ ($C_{645}$)

Order 15: $C_{15}$ ($C_{774}$)

Order 10: $C_{10}$ ($C_{1161}$)

Order 9: $C_9$ ($C_{1290}$)

Order 6: $C_6$ ($C_{1935}$)

Order 5: $C_5$ ($C_{2322}$)

Order 3: $C_3$ ($C_{3870}$)

Order 2: $C_2$ ($C_{5805}$)

Order 1: $C_1$ ($C_{11610}$)

Series

Derived series

$C_{11610}$

$\rhd$

$C_1$

Chief series

$C_{11610}$

$\rhd$

$C_{5805}$

$\rhd$

$C_{1935}$

$\rhd$

$C_{645}$

$\rhd$

$C_{215}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Lower central series

$C_{11610}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{11610}$

Supergroups

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

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

Character theory

Complex character table

The $11610 \times 11610$ character table is not available for this group.

Rational character table

The $32 \times 32$ rational character table is not available for this group.