Abstract group $C(2,3)$

• Label: 25920.a
• Order: \( 2^{6} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{7} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $2$

Group information

Description:	$C(2,3)$
Order:	\(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$\SO(5,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \)
Outer automorphisms:	$C_2$, of order \(2\)
Composition factors:	$C(2,3)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	9	12
Elements	1	315	800	3780	5184	5760	5760	4320	25920
Conjugacy classes	1	2	4	2	1	6	2	2	20
Divisions	1	2	3	2	1	4	1	1	15
Autjugacy classes	1	2	3	2	1	4	1	1	15

Dimension	1	5	6	10	15	20	24	30	40	45	60	64	80	81	90
Irr. complex chars.	1	2	1	2	2	1	1	3	2	2	1	1	0	1	0	20
Irr. rational chars.	1	0	1	1	2	2	1	1	0	0	2	1	1	1	1	15

Minimal Presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$2$
Inequivalent generating pairs:	$11505$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	5	6	6
Arbitrary	5	6	6

Constructions

Groups of Lie type:	$\SU(4,2)$, $\Omega(5,3)$, $\OmegaMinus(6,2)$, $\SpinMinus(6,2)$, $\PSp(4,3)$, $\PGU(4,2)$
Permutation group:	Degree $27$ $\langle(1,2,20,25)(3,17,15,22)(4,8)(5,18,24,7)(6,19,26,9)(10,11,27,21)(13,14)(16,23) \!\cdots\! \rangle$
Transitive group:	27T993	36T12781	40T14344	40T14345	all 5
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SU(4,2)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 45649 subgroups in 116 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C(2,3)$
Commutator:	$G' \simeq$ $C(2,3)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C(2,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $C(2,3)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $C(2,3)$
Socle:	$\operatorname{soc} \simeq$ $C(2,3)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 25920: $C(2,3)$

Order 960: $C_2^4:A_5$

Order 720: $S_6$

Order 648: $C_3^3:S_4$, $\GU(3,2)$

Order 576: $\OmegaPlus(4,3):C_2$

Order 360: $A_6$

Order 324: $C_3^3:A_4$

Order 288: $\OmegaPlus(4,3)$

Order 216: $\SU(3,2)$, $S_3^2:S_3$

Order 192: $C_2\wr A_4$, $C_2^3:S_4$

Order 162: $C_3\wr S_3$

Order 160: $C_2^4:D_5$

Order 120: $S_5$ x 2

Order 108: $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^3:C_4$, $\He_3:C_4$

Order 96: $Q_8:A_4$ x 2, $\SL(2,3):C_2^2$, $\GL(2,\mathbb{Z}/4)$

Order 81: $C_3\wr C_3$

Order 80: $C_2^4:C_5$

Order 72: $\SOPlus(4,2)$, $C_3\times \SL(2,3)$

Order 64: $C_2\wr C_2^2$

Order 60: $A_5$ x 2

Order 54: $C_3^2:S_3$, $C_3^2:C_6$, $S_3\times C_3^2$

Order 48: $C_2\times S_4$ x 2, $C_2^2\times A_4$, $\SL(2,3):C_2$, $A_4:C_4$

Order 36: $S_3^2$ x 2, $C_3^2:C_4$, $C_6\times S_3$

Order 32: $C_2^3:C_4$, $D_4:C_2^2$, $C_2^2\wr C_2$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $\SL(2,3)$ x 3, $C_2\times A_4$ x 3, $S_4$ x 3, $C_3:D_4$, $C_3\times Q_8$

Order 20: $F_5$

Order 18: $C_3\times S_3$ x 5, $C_3\times C_6$, $C_3:S_3$

Order 16: $C_2^2:C_4$ x 2, $C_2\times D_4$ x 2, $C_2^4$, $D_4:C_2$

Order 12: $D_6$ x 2, $A_4$ x 2, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 10: $D_5$

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

Order 8: $C_2^3$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 4, $S_3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 4, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 25920: $C(2,3)$

Order 960: $C_2^4:A_5$

Order 720: $S_6$

Order 648: $C_3^3:S_4$, $\GU(3,2)$

Order 576: $\OmegaPlus(4,3):C_2$

Order 360: $A_6$

Order 324: $C_3^3:A_4$

Order 288: $\OmegaPlus(4,3)$

Order 216: $\SU(3,2)$, $S_3^2:S_3$

Order 192: $C_2\wr A_4$, $C_2^3:S_4$

Order 162: $C_3\wr S_3$

Order 160: $C_2^4:D_5$

Order 120: $S_5$ x 2

Order 108: $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^3:C_4$, $\He_3:C_4$

Order 96: $Q_8:A_4$ x 2, $\SL(2,3):C_2^2$, $\GL(2,\mathbb{Z}/4)$

Order 81: $C_3\wr C_3$

Order 80: $C_2^4:C_5$

Order 72: $\SOPlus(4,2)$, $C_3\times \SL(2,3)$

Order 64: $C_2\wr C_2^2$

Order 60: $A_5$ x 2

Order 54: $C_3^2:S_3$, $C_3^2:C_6$, $S_3\times C_3^2$

Order 48: $C_2\times S_4$ x 2, $C_2^2\times A_4$, $\SL(2,3):C_2$, $A_4:C_4$

Order 36: $S_3^2$ x 2, $C_3^2:C_4$, $C_6\times S_3$

Order 32: $C_2^3:C_4$, $D_4:C_2^2$, $C_2^2\wr C_2$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $\SL(2,3)$ x 3, $C_2\times A_4$ x 3, $S_4$ x 3, $C_3:D_4$, $C_3\times Q_8$

Order 20: $F_5$

Order 18: $C_3\times S_3$ x 5, $C_3\times C_6$, $C_3:S_3$

Order 16: $C_2^2:C_4$ x 2, $C_2\times D_4$ x 2, $C_2^4$, $D_4:C_2$

Order 12: $D_6$ x 2, $A_4$ x 2, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 10: $D_5$

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

Order 8: $C_2^3$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 4, $S_3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 4, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 25920: $C(2,3)$ ($C_1$)

Order 1: $C_1$ ($C(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 25920: $C(2,3)$ ($C_1$)

Order 1: $C_1$ ($C(2,3)$)

Series

Derived series	$C(2,3)$
Chief series	$C(2,3)$	$\rhd$	$C_1$
Lower central series	$C(2,3)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

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

Character theory

Complex character table

See the $20 \times 20$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	3A	3B	3C	4A	4B	5A	6A	6B	6C	6D	9A	12A
	Size	1	45	270	80	240	480	540	3240	5184	720	1440	1440	2160	5760	4320
	2 P	1A	1A	1A	3A	3B	3C	2A	2B	5A	3A	3B	3C	3B	9A	6A
	3 P	1A	2A	2B	1A	1A	1A	4A	4B	5A	2A	2A	2A	2B	3A	4A
	5 P	1A	2A	2B	3A	3B	3C	4A	4B	1A	6A	6B	6C	6D	9A	12A
25920.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
25920.a.5a		$10$	$-6$	$2$	$1$	$-2$	$4$	$2$	$-2$	$0$	$-3$	$0$	$0$	$2$	$1$	$-1$
25920.a.6a		$6$	$-2$	$2$	$-3$	$3$	$0$	$2$	$0$	$1$	$1$	$1$	$-2$	$-1$	$0$	$-1$
25920.a.10a		$20$	$4$	$-4$	$-7$	$2$	$2$	$4$	$0$	$0$	$1$	$-2$	$-2$	$2$	$-1$	$1$
25920.a.15a		$15$	$7$	$3$	$-3$	$0$	$3$	$-1$	$1$	$0$	$1$	$-2$	$1$	$0$	$0$	$-1$
25920.a.15b		$15$	$-1$	$-1$	$6$	$3$	$0$	$3$	$-1$	$0$	$2$	$-1$	$2$	$-1$	$0$	$0$
25920.a.20a		$20$	$4$	$4$	$2$	$5$	$-1$	$0$	$0$	$0$	$-2$	$1$	$1$	$1$	$-1$	$0$
25920.a.24a		$24$	$8$	$0$	$6$	$0$	$3$	$0$	$0$	$-1$	$2$	$2$	$-1$	$0$	$0$	$0$
25920.a.30a		$30$	$-10$	$2$	$3$	$3$	$3$	$-2$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$0$	$1$
25920.a.30b		$60$	$12$	$4$	$-3$	$-6$	$0$	$4$	$0$	$0$	$-3$	$0$	$0$	$-2$	$0$	$1$
25920.a.40a		$80$	$-16$	$0$	$-10$	$-4$	$2$	$0$	$0$	$0$	$2$	$2$	$2$	$0$	$-1$	$0$
25920.a.45a		$90$	$-6$	$-6$	$9$	$0$	$0$	$2$	$2$	$0$	$-3$	$0$	$0$	$0$	$0$	$-1$
25920.a.60a		$60$	$-4$	$4$	$6$	$-3$	$-3$	$0$	$0$	$0$	$2$	$-1$	$-1$	$1$	$0$	$0$
25920.a.64a		$64$	$0$	$0$	$-8$	$4$	$-2$	$0$	$0$	$-1$	$0$	$0$	$0$	$0$	$1$	$0$
25920.a.81a		$81$	$9$	$-3$	$0$	$0$	$0$	$-3$	$-1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$