Abstract group 9288.a: $C_{9288}$

• Label: 9288.a
• Order: \( 2^{3} \cdot 3^{3} \cdot 43 \)
• Exponent: \( 2^{3} \cdot 3^{3} \cdot 43 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 7 \)
• Perm deg.: $78$
• Trans deg.: $9288$
• Rank: $1$

Group information

Description:	$C_{9288}$
Order:	\(9288\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 43 \)
Exponent:	\(9288\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 43 \)
Automorphism group:	$C_2^2\times C_6\times C_{126}$, of order \(3024\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 3, $C_3$ x 3, $C_{43}$
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	3	4	6	8	9	12	18	24	27	36	43	54	72	86	108	129	172	216	258	344	387	516	774	1032	1161	1548	2322	3096	4644	9288
Elements	1	1	2	2	2	4	6	4	6	8	18	12	42	18	24	42	36	84	84	72	84	168	252	168	252	336	756	504	756	1008	1512	3024	9288
Conjugacy classes	1	1	2	2	2	4	6	4	6	8	18	12	42	18	24	42	36	84	84	72	84	168	252	168	252	336	756	504	756	1008	1512	3024	9288
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:	$78$
Transitive degree:	$9288$
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:	$\langle a \mid a^{9288}=1 \rangle$
Permutation group:	Degree $78$ $\langle(1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 154 & 17 \\ 64 & 154 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{431})$
Direct product:	$C_8$ $\, \times\, $ $C_{27}$ $\, \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:	$C_{4644}$ . $C_2$	$C_{3096}$ . $C_3$	$C_{2322}$ . $C_4$	$C_{1548}$ . $C_6$	all 24

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

Homology

Primary decomposition:	$C_{8} \times C_{27} \times C_{43}$
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:	$Z \simeq$ $C_{9288}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{9288}$
Frattini:	$\Phi \simeq$ $C_{36}$	$G/\Phi \simeq$ $C_{258}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{9288}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{9288}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{258}$	$G/\operatorname{soc} \simeq$ $C_{36}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_{27}$
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 9288: $C_{9288}$

Order 4644: $C_{4644}$

Order 3096: $C_{3096}$

Order 2322: $C_{2322}$

Order 1548: $C_{1548}$

Order 1161: $C_{1161}$

Order 1032: $C_{1032}$

Order 774: $C_{774}$

Order 516: $C_{516}$

Order 387: $C_{387}$

Order 344: $C_{344}$

Order 258: $C_{258}$

Order 216: $C_{216}$

Order 172: $C_{172}$

Order 129: $C_{129}$

Order 108: $C_{108}$

Order 86: $C_{86}$

Order 72: $C_{72}$

Order 54: $C_{54}$

Order 43: $C_{43}$

Order 36: $C_{36}$

Order 27: $C_{27}$

Order 24: $C_{24}$

Order 18: $C_{18}$

Order 12: $C_{12}$

Order 9: $C_9$

Order 8: $C_8$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9288: $C_{9288}$

Order 4644: $C_{4644}$

Order 3096: $C_{3096}$

Order 2322: $C_{2322}$

Order 1548: $C_{1548}$

Order 1161: $C_{1161}$

Order 1032: $C_{1032}$

Order 774: $C_{774}$

Order 516: $C_{516}$

Order 387: $C_{387}$

Order 344: $C_{344}$

Order 258: $C_{258}$

Order 216: $C_{216}$

Order 172: $C_{172}$

Order 129: $C_{129}$

Order 108: $C_{108}$

Order 86: $C_{86}$

Order 72: $C_{72}$

Order 54: $C_{54}$

Order 43: $C_{43}$

Order 36: $C_{36}$

Order 27: $C_{27}$

Order 24: $C_{24}$

Order 18: $C_{18}$

Order 12: $C_{12}$

Order 9: $C_9$

Order 8: $C_8$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9288: $C_{9288}$ ($C_1$)

Order 4644: $C_{4644}$ ($C_2$)

Order 3096: $C_{3096}$ ($C_3$)

Order 2322: $C_{2322}$ ($C_4$)

Order 1548: $C_{1548}$ ($C_6$)

Order 1161: $C_{1161}$ ($C_8$)

Order 1032: $C_{1032}$ ($C_9$)

Order 774: $C_{774}$ ($C_{12}$)

Order 516: $C_{516}$ ($C_{18}$)

Order 387: $C_{387}$ ($C_{24}$)

Order 344: $C_{344}$ ($C_{27}$)

Order 258: $C_{258}$ ($C_{36}$)

Order 216: $C_{216}$ ($C_{43}$)

Order 172: $C_{172}$ ($C_{54}$)

Order 129: $C_{129}$ ($C_{72}$)

Order 108: $C_{108}$ ($C_{86}$)

Order 86: $C_{86}$ ($C_{108}$)

Order 72: $C_{72}$ ($C_{129}$)

Order 54: $C_{54}$ ($C_{172}$)

Order 43: $C_{43}$ ($C_{216}$)

Order 36: $C_{36}$ ($C_{258}$)

Order 27: $C_{27}$ ($C_{344}$)

Order 24: $C_{24}$ ($C_{387}$)

Order 18: $C_{18}$ ($C_{516}$)

Order 12: $C_{12}$ ($C_{774}$)

Order 9: $C_9$ ($C_{1032}$)

Order 8: $C_8$ ($C_{1161}$)

Order 6: $C_6$ ($C_{1548}$)

Order 4: $C_4$ ($C_{2322}$)

Order 3: $C_3$ ($C_{3096}$)

Order 2: $C_2$ ($C_{4644}$)

Order 1: $C_1$ ($C_{9288}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9288: $C_{9288}$ ($C_1$)

Order 4644: $C_{4644}$ ($C_2$)

Order 3096: $C_{3096}$ ($C_3$)

Order 2322: $C_{2322}$ ($C_4$)

Order 1548: $C_{1548}$ ($C_6$)

Order 1161: $C_{1161}$ ($C_8$)

Order 1032: $C_{1032}$ ($C_9$)

Order 774: $C_{774}$ ($C_{12}$)

Order 516: $C_{516}$ ($C_{18}$)

Order 387: $C_{387}$ ($C_{24}$)

Order 344: $C_{344}$ ($C_{27}$)

Order 258: $C_{258}$ ($C_{36}$)

Order 216: $C_{216}$ ($C_{43}$)

Order 172: $C_{172}$ ($C_{54}$)

Order 129: $C_{129}$ ($C_{72}$)

Order 108: $C_{108}$ ($C_{86}$)

Order 86: $C_{86}$ ($C_{108}$)

Order 72: $C_{72}$ ($C_{129}$)

Order 54: $C_{54}$ ($C_{172}$)

Order 43: $C_{43}$ ($C_{216}$)

Order 36: $C_{36}$ ($C_{258}$)

Order 27: $C_{27}$ ($C_{344}$)

Order 24: $C_{24}$ ($C_{387}$)

Order 18: $C_{18}$ ($C_{516}$)

Order 12: $C_{12}$ ($C_{774}$)

Order 9: $C_9$ ($C_{1032}$)

Order 8: $C_8$ ($C_{1161}$)

Order 6: $C_6$ ($C_{1548}$)

Order 4: $C_4$ ($C_{2322}$)

Order 3: $C_3$ ($C_{3096}$)

Order 2: $C_2$ ($C_{4644}$)

Order 1: $C_1$ ($C_{9288}$)

Series

Derived series

$C_{9288}$

$\rhd$

$C_1$

Chief series

$C_{9288}$

$\rhd$

$C_{4644}$

$\rhd$

$C_{2322}$

$\rhd$

$C_{1161}$

$\rhd$

$C_{387}$

$\rhd$

$C_{129}$

$\rhd$

$C_{43}$

$\rhd$

$C_1$

Lower central series

$C_{9288}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{9288}$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $9288 \times 9288$ character table is not available for this group.

Rational character table

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