Abstract group 1944.2330: $C_3^3:\PSU(3,2)$

• Label: 1944.2330
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $2$

Group information

Description:	$C_3^3:\PSU(3,2)$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4:(S_3\times \GL(2,3))$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	81	242	486	162	972	1944
Conjugacy classes	1	1	8	3	2	6	21
Divisions	1	1	6	3	1	3	15
Autjugacy classes	1	1	5	1	1	1	10

Dimension	1	2	4	8	16	24
Irr. complex chars.	12	3	0	3	0	3	21
Irr. rational chars.	4	5	1	1	1	3	15

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	24	24	24

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{12}=c^{3}=d^{3}=e^{3}=f^{3}=[c,d]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(2,5,3,9)(4,8,7,6)(11,14,12,18)(13,17,16,15)(20,23,21,27)(22,26,25,24), (2,8,3,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 0 & 1 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 2 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 2 & 2 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 2 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 2 & 2 \\ 0 & 1 & 0 & 0 \\ 2 & 1 & 1 & 2 \\ 1 & 1 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	27T415	36T2813	36T2814	36T2994	all 5
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $\PSU(3,2)$	$(C_3^4:Q_8)$ $\,\rtimes\,$ $C_3$	$(C_3^4:C_3)$ $\,\rtimes\,$ $Q_8$	$C_3^4$ $\,\rtimes\,$ $(C_3\times Q_8)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^4:C_4)$ . $C_6$ (3)	$(C_3^4:C_{12})$ . $C_2$ (3)	$C_3^2$ . $(C_3^3:Q_8)$	$(C_3^4:C_6)$ . $C_2^2$	all 5

Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 4300 subgroups in 109 conjugacy classes, 15 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3:\PSU(3,2)$
Commutator:	$G' \simeq$ $C_3^3:S_3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^3:Q_8$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4:C_3$	$G/\operatorname{Fit} \simeq$ $Q_8$
Radical:	$R \simeq$ $C_3^3:\PSU(3,2)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $C_3^3:Q_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$

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 1944: $C_3^3:\PSU(3,2)$

Order 972: $C_3^4:C_{12}$ x 3

Order 648: $C_3^4:Q_8$

Order 486: $C_3^4:C_6$

Order 324: $C_3^4:C_4$ x 3

Order 243: $C_3^4:C_3$

Order 216: $C_3^3:Q_8$

Order 162: $C_3^3:S_3$, $C_3^3:C_6$

Order 108: $C_3^2:C_{12}$ x 3

Order 81: $C_3\times \He_3$ x 2, $C_3^4$

Order 72: $\PSU(3,2)$ x 2

Order 54: $C_3^2:S_3$ x 4, $C_3^2:C_6$ x 3, $C_3^2:C_6$

Order 36: $C_3^2:C_4$ x 9

Order 27: $C_3^3$ x 6, $\He_3$ x 6

Order 24: $C_3\times Q_8$

Order 18: $C_3:S_3$ x 16, $C_3\times S_3$

Order 12: $C_{12}$ x 3

Order 9: $C_3^2$ x 21

Order 8: $Q_8$

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

Order 4: $C_4$ x 3

Order 3: $C_3$ x 6

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_3^3:\PSU(3,2)$

Order 972: $C_3^4:C_{12}$

Order 648: $C_3^4:Q_8$

Order 486: $C_3^4:C_6$

Order 324: $C_3^4:C_4$

Order 243: $C_3^4:C_3$

Order 216: $C_3^3:Q_8$

Order 162: $C_3^3:S_3$, $C_3^3:C_6$

Order 108: $C_3^2:C_{12}$

Order 81: $C_3\times \He_3$ x 2, $C_3^4$

Order 72: $\PSU(3,2)$ x 2

Order 54: $C_3^2:S_3$ x 3, $C_3^2:C_6$ x 2, $C_3^2:C_6$

Order 36: $C_3^2:C_4$ x 3

Order 27: $C_3^3$ x 5, $\He_3$ x 4

Order 24: $C_3\times Q_8$

Order 18: $C_3:S_3$ x 8, $C_3\times S_3$

Order 12: $C_{12}$

Order 9: $C_3^2$ x 11

Order 8: $Q_8$

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

Order 4: $C_4$

Order 3: $C_3$ x 5

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_3^3:\PSU(3,2)$ ($C_1$)

Order 972: $C_3^4:C_{12}$ ($C_2$) x 3

Order 648: $C_3^4:Q_8$ ($C_3$)

Order 486: $C_3^4:C_6$ ($C_2^2$)

Order 324: $C_3^4:C_4$ ($C_6$) x 3

Order 243: $C_3^4:C_3$ ($Q_8$)

Order 162: $C_3^3:S_3$ ($C_2\times C_6$)

Order 81: $C_3^4$ ($C_3\times Q_8$)

Order 27: $C_3^3$ ($\PSU(3,2)$)

Order 9: $C_3^2$ ($C_3^3:Q_8$)

Order 1: $C_1$ ($C_3^3:\PSU(3,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_3^3:\PSU(3,2)$ ($C_1$)

Order 972: $C_3^4:C_{12}$ ($C_2$)

Order 648: $C_3^4:Q_8$ ($C_3$)

Order 486: $C_3^4:C_6$ ($C_2^2$)

Order 324: $C_3^4:C_4$ ($C_6$)

Order 243: $C_3^4:C_3$ ($Q_8$)

Order 162: $C_3^3:S_3$ ($C_2\times C_6$)

Order 81: $C_3^4$ ($C_3\times Q_8$)

Order 27: $C_3^3$ ($\PSU(3,2)$)

Order 9: $C_3^2$ ($C_3^3:Q_8$)

Order 1: $C_1$ ($C_3^3:\PSU(3,2)$)

Series

Derived series

$C_3^3:\PSU(3,2)$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^3:\PSU(3,2)$

$\rhd$

$C_3^4:C_{12}$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_3^3:\PSU(3,2)$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 7 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 $21 \times 21$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	3B	3C	3D	3E	3F	4A	4B	4C	6A	12A	12B	12C
	Size	1	81	8	18	24	24	24	144	162	162	162	162	324	324	324
	2 P	1A	1A	3A	3B	3C	3D	3E	3F	2A	2A	2A	3B	6A	6A	6A
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	4A	4B	4C	2A	4A	4B	4C
	Schur
1944.2330.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
1944.2330.1b		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$
1944.2330.1c		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
1944.2330.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$
1944.2330.1e		$2$	$2$	$2$	$-1$	$2$	$2$	$2$	$-1$	$2$	$2$	$2$	$-1$	$-1$	$-1$	$-1$
1944.2330.1f		$2$	$2$	$2$	$-1$	$2$	$2$	$2$	$-1$	$-2$	$-2$	$2$	$-1$	$1$	$1$	$-1$
1944.2330.1g		$2$	$2$	$2$	$-1$	$2$	$2$	$2$	$-1$	$-2$	$2$	$-2$	$-1$	$1$	$-1$	$1$
1944.2330.1h		$2$	$2$	$2$	$-1$	$2$	$2$	$2$	$-1$	$2$	$-2$	$-2$	$-1$	$-1$	$1$	$1$
1944.2330.2a	2	$2$	$-2$	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$
1944.2330.2b		$4$	$-4$	$4$	$-2$	$4$	$4$	$4$	$-2$	$0$	$0$	$0$	$2$	$0$	$0$	$0$
1944.2330.8a		$8$	$0$	$8$	$8$	$-1$	$-1$	$-1$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
1944.2330.8b		$16$	$0$	$16$	$-8$	$-2$	$-2$	$-2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
1944.2330.24a		$24$	$0$	$-3$	$0$	$6$	$-3$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
1944.2330.24b		$24$	$0$	$-3$	$0$	$-3$	$-3$	$6$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
1944.2330.24c		$24$	$0$	$-3$	$0$	$-3$	$6$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$