Abstract group 5184.td: $C_3\times C_6^2:\GL(2,3)$

• Label: 5184.td
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $16$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_3\times C_6^2:\GL(2,3)$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_6^2:(D_6\times \GL(2,3))$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4
Derived length:	$5$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	183	242	216	2814	432	432	864	5184
Conjugacy classes	1	11	11	4	73	8	8	16	132
Divisions	1	11	7	4	45	4	4	4	80
Autjugacy classes	1	4	7	2	16	2	2	2	36

Dimension	1	2	3	4	6	8	16	32
Irr. complex chars.	24	36	24	12	0	24	12	0	132
Irr. rational chars.	8	12	8	12	8	16	12	4	80

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$1747200$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=d^{12}=e^{3}=f^{6}=[a,f]=[b,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(3,6)(5,9)(7,8), (1,2,4)(3,5,8)(6,9,7), (1,3,6)(2,5,9)(4,8,7), (13,14), (2,3,7)(4,6,5), (10,11,12)(13,14)(15,16)\rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $(C_3^2:\GL(2,3))$
Semidirect product:	$(C_3\times C_6^2:Q_8)$ $\,\rtimes\,$ $S_3$	$(C_6\times \PSU(3,2))$ $\,\rtimes\,$ $D_6$	$\PSU(3,2)$ $\,\rtimes\,$ $(C_6\times D_6)$	$(C_6^2:\SL(2,3))$ $\,\rtimes\,$ $C_6$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2:C_6)$ . $S_4$	$(C_6:S_3)$ . $(C_6\times S_4)$	$(C_6:D_6)$ . $(C_3\times S_4)$	$(C_3^2:D_6)$ . $(C_2\times S_4)$	all 6
Aut. group:	$\Aut(C_3^2:C_{28})$	$\Aut(C_{42}:S_3)$	$\Aut(C_3^2:C_{36})$	$\Aut(C_3:S_3\times C_{18})$	all 13

Elements of the group are displayed as permutations of degree 16.

Homology

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

Subgroups

There are 25504 subgroups in 1353 conjugacy classes, 84 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_3^2:\GL(2,3)$
Commutator:	$G' \simeq$ $\PU(3,2)$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3\times C_6^2:\GL(2,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $\GL(2,3)$
Radical:	$R \simeq$ $C_3\times C_6^2:\GL(2,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $\GL(2,3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times \He_3$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 5184: $C_3\times C_6^2:\GL(2,3)$

Order 2592: $C_6\times C_3^2:\GL(2,3)$ x 6, $C_2\times C_6\times \ASL(2,3)$

Order 1728: $C_2^2\times F_9:C_6$, $C_6^2:\GL(2,3)$

Order 1296: $C_3^3:\GL(2,3)$ x 4, $C_6\times \PU(3,2)$ x 3, $C_6^2.S_3^2$

Order 864: $C_2\times F_9:C_6$ x 12, $C_2\times C_3^2:\GL(2,3)$ x 6, $C_6^2:\SL(2,3)$ x 2, $C_3\times C_6^2:Q_8$, $D_6^2:C_6$, $C_6^2:C_{24}$

Order 648: $C_6\times C_3^2:D_6$ x 12, $C_3\times C_6^2:C_6$ x 2, $C_3^3:\SL(2,3)$, $C_3\times C_6^2:S_3$

Order 576: $F_9:C_2^3$, $C_2\times C_6\times \GL(2,3)$

Order 432: $F_9:C_6$ x 16, $C_6\times \SOPlus(4,2)$ x 9, $C_6\times \PSU(3,2)$ x 6, $C_6\times F_9$ x 6, $C_2\times \PU(3,2)$ x 6, $C_3^2:\GL(2,3)$ x 4, $C_3\times D_6^2$ x 2, $C_6^2:C_{12}$, $C_6^2:D_6$

Order 324: $C_3^3:D_6$ x 16, $C_3^2:C_6^2$ x 12, $C_3^3:D_6$ x 6, $C_3^2:C_6^2$

Order 288: $F_9:C_2^2$ x 12, $C_6\times \GL(2,3)$ x 6, $Q_8:C_6^2$, $C_6^2:Q_8$, $C_6^2:D_4$, $C_2^2\times F_9$

Order 216: $C_6\times S_3^2$ x 21, $C_6.S_3^2$ x 12, $S_3^2:C_6$ x 10, $C_2\times C_3^2:C_{12}$ x 6, $C_3^3:Q_8$ x 5, $S_3\times C_6^2$ x 4, $C_3\times F_9$ x 4, $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 2, $\PU(3,2)$ x 2, $C_6^2:S_3$

Order 192: $C_2^2\times \GL(2,3)$, $C_{24}:C_2^3$

Order 162: $C_3^3:C_6$ x 8, $C_3^3:S_3$ x 4, $C_6\times \He_3$ x 3

Order 144: $F_9:C_2$ x 16, $S_3^2:C_2^2$ x 9, $C_2\times \PSU(3,2)$ x 6, $C_2\times F_9$ x 6, $C_3\times \GL(2,3)$ x 4, $C_6\times \SL(2,3)$ x 3, $C_6^2:C_2^2$ x 2, $D_6^2$ x 2, $C_6^2:C_4$

Order 108: $C_3\times S_3^2$ x 26, $C_3^2\times D_6$ x 24, $C_3^2:D_6$ x 24, $C_3^2:D_6$ x 16, $C_3^2:D_6$ x 12, $C_3^2:D_6$ x 6, $C_3^2:C_{12}$ x 4, $C_2^2\times \He_3$ x 4, $C_3\times C_6^2$ x 3

Order 96: $C_6\times \SD_{16}$ x 12, $C_2\times \GL(2,3)$ x 6, $C_2^2\times \SL(2,3)$ x 2, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_2^2\times C_{24}$

Order 81: $C_3\times \He_3$

Order 72: $C_6\times D_6$ x 38, $S_3\times D_6$ x 21, $\SOPlus(4,2)$ x 10, $C_2\times C_3^2:C_4$ x 6, $\PSU(3,2)$ x 5, $F_9$ x 4, $C_2\times C_6^2$ x 2, $C_6:D_6$ x 2, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$

Order 54: $C_3^2:C_6$ x 16, $S_3\times C_3^2$ x 16, $C_2\times \He_3$ x 12, $C_3^2\times C_6$ x 9, $C_3^2:C_6$ x 8, $C_3^2:S_3$ x 4

Order 48: $C_3\times \SD_{16}$ x 16, $C_6\times D_4$ x 9, $C_6\times Q_8$ x 6, $C_2\times \SL(2,3)$ x 6, $C_2\times C_{24}$ x 6, $\GL(2,3)$ x 4, $C_2^2\times D_6$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$

Order 36: $C_6\times S_3$ x 116, $S_3^2$ x 26, $C_6^2$ x 24, $C_6:S_3$ x 12, $C_3^2:C_4$ x 4

Order 32: $C_2\times \SD_{16}$ x 12, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

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

Order 24: $C_2\times D_6$ x 29, $C_2^2\times C_6$ x 15, $C_3\times D_4$ x 10, $C_2\times C_{12}$ x 6, $C_3\times Q_8$ x 5, $C_{24}$ x 4, $\SL(2,3)$ x 2

Order 18: $C_3\times S_3$ x 56, $C_3\times C_6$ x 44, $C_3:S_3$ x 8

Order 16: $\SD_{16}$ x 16, $C_2\times D_4$ x 9, $C_2\times C_8$ x 6, $C_2\times Q_8$ x 6, $C_2^4$, $C_2^2\times C_4$

Order 12: $D_6$ x 62, $C_2\times C_6$ x 53, $C_{12}$ x 4

Order 9: $C_3^2$ x 12

Order 8: $C_2^3$ x 11, $D_4$ x 10, $C_2\times C_4$ x 6, $Q_8$ x 5, $C_8$ x 4

Order 6: $C_6$ x 45, $S_3$ x 20

Order 4: $C_2^2$ x 23, $C_4$ x 4

Order 3: $C_3$ x 7

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $C_3\times C_6^2:\GL(2,3)$

Order 2592: $C_2\times C_6\times \ASL(2,3)$, $C_6\times C_3^2:\GL(2,3)$

Order 1728: $C_2^2\times F_9:C_6$, $C_6^2:\GL(2,3)$

Order 1296: $C_6\times \PU(3,2)$, $C_3^3:\GL(2,3)$, $C_6^2.S_3^2$

Order 864: $C_2\times F_9:C_6$ x 2, $C_6^2:\SL(2,3)$ x 2, $C_3\times C_6^2:Q_8$, $D_6^2:C_6$, $C_6^2:C_{24}$, $C_2\times C_3^2:\GL(2,3)$

Order 648: $C_3\times C_6^2:C_6$ x 2, $C_6\times C_3^2:D_6$ x 2, $C_3^3:\SL(2,3)$, $C_3\times C_6^2:S_3$

Order 576: $F_9:C_2^3$, $C_2\times C_6\times \GL(2,3)$

Order 432: $C_3\times D_6^2$ x 2, $C_6\times \PSU(3,2)$ x 2, $C_6\times \SOPlus(4,2)$ x 2, $F_9:C_6$ x 2, $C_2\times \PU(3,2)$ x 2, $C_6^2:C_{12}$, $C_6\times F_9$, $C_3^2:\GL(2,3)$, $C_6^2:D_6$

Order 324: $C_3^2:C_6^2$ x 3, $C_3^3:D_6$ x 2, $C_3^2:C_6^2$, $C_3^3:D_6$

Order 288: $F_9:C_2^2$ x 2, $Q_8:C_6^2$, $C_6\times \GL(2,3)$, $C_6^2:Q_8$, $C_6^2:D_4$, $C_2^2\times F_9$

Order 216: $S_3\times C_6^2$ x 4, $C_6\times S_3^2$ x 4, $C_6^2:C_6$ x 4, $C_3^3:Q_8$ x 3, $C_6^2:C_6$ x 2, $C_2\times C_3^2:C_{12}$ x 2, $S_3^2:C_6$ x 2, $\PU(3,2)$ x 2, $C_6.S_3^2$ x 2, $C_3\times F_9$, $C_6^2:S_3$

Order 192: $C_2^2\times \GL(2,3)$, $C_{24}:C_2^3$

Order 162: $C_3^3:C_6$ x 3, $C_6\times \He_3$, $C_3^3:S_3$

Order 144: $C_6^2:C_2^2$ x 2, $D_6^2$ x 2, $C_2\times \PSU(3,2)$ x 2, $S_3^2:C_2^2$ x 2, $F_9:C_2$ x 2, $C_6^2:C_4$, $C_2\times F_9$, $C_6\times \SL(2,3)$, $C_3\times \GL(2,3)$

Order 108: $C_3^2:D_6$ x 6, $C_3^2\times D_6$ x 5, $C_3\times S_3^2$ x 4, $C_2^2\times \He_3$ x 4, $C_3\times C_6^2$ x 3, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$

Order 96: $C_2^2\times \SL(2,3)$ x 2, $C_6\times \SD_{16}$ x 2, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_2\times \GL(2,3)$, $C_2^2\times C_{24}$

Order 81: $C_3\times \He_3$

Order 72: $C_6\times D_6$ x 17, $S_3\times D_6$ x 4, $\PSU(3,2)$ x 3, $C_2\times C_6^2$ x 2, $C_6:D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $\SOPlus(4,2)$ x 2, $F_9$, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 5, $C_2\times \He_3$ x 4, $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 3, $C_3^2:S_3$

Order 48: $C_2^2\times D_6$ x 2, $C_6\times Q_8$ x 2, $C_6\times D_4$ x 2, $C_2\times \SL(2,3)$ x 2, $C_3\times \SD_{16}$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $\GL(2,3)$, $C_2\times C_{24}$

Order 36: $C_6\times S_3$ x 20, $C_6^2$ x 14, $S_3^2$ x 4, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2

Order 32: $C_2\times \SD_{16}$ x 2, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

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

Order 24: $C_2\times D_6$ x 9, $C_2^2\times C_6$ x 8, $C_3\times Q_8$ x 3, $C_2\times C_{12}$ x 2, $\SL(2,3)$ x 2, $C_3\times D_4$ x 2, $C_{24}$

Order 18: $C_3\times S_3$ x 16, $C_3\times C_6$ x 14, $C_3:S_3$ x 3

Order 16: $\SD_{16}$ x 2, $C_2\times Q_8$ x 2, $C_2\times D_4$ x 2, $C_2\times C_8$, $C_2^4$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 18, $D_6$ x 10, $C_{12}$ x 2

Order 9: $C_3^2$ x 11

Order 8: $C_2^3$ x 4, $Q_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $C_8$

Order 6: $C_6$ x 16, $S_3$ x 6

Order 4: $C_2^2$ x 6, $C_4$ x 2

Order 3: $C_3$ x 7

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $C_3\times C_6^2:\GL(2,3)$ ($C_1$)

Order 2592: $C_6\times C_3^2:\GL(2,3)$ ($C_2$) x 6, $C_2\times C_6\times \ASL(2,3)$ ($C_2$)

Order 1728: $C_6^2:\GL(2,3)$ ($C_3$)

Order 1296: $C_3^3:\GL(2,3)$ ($C_2^2$) x 4, $C_6\times \PU(3,2)$ ($C_2^2$) x 3

Order 864: $C_2\times C_3^2:\GL(2,3)$ ($C_6$) x 6, $C_3\times C_6^2:Q_8$ ($S_3$), $C_6^2:\SL(2,3)$ ($C_6$)

Order 648: $C_3^3:\SL(2,3)$ ($C_2^3$)

Order 432: $C_3^2:\GL(2,3)$ ($C_2\times C_6$) x 4, $C_6\times \PSU(3,2)$ ($D_6$) x 3, $C_2\times \PU(3,2)$ ($C_2\times C_6$) x 3

Order 288: $C_6^2:Q_8$ ($C_3\times S_3$)

Order 216: $C_6^2:C_6$ ($S_4$), $C_3^3:Q_8$ ($C_2\times D_6$), $\PU(3,2)$ ($C_2^2\times C_6$)

Order 144: $C_2\times \PSU(3,2)$ ($C_6\times S_3$) x 3

Order 108: $C_3^2:D_6$ ($C_2\times S_4$) x 3, $C_3^2:D_6$ ($\GL(2,3)$) x 3, $C_3\times C_6^2$ ($\GL(2,3)$)

Order 72: $C_6:D_6$ ($C_3\times S_4$), $\PSU(3,2)$ ($C_6\times D_6$)

Order 54: $C_3^2\times C_6$ ($C_2\times \GL(2,3)$) x 3, $C_3^2:C_6$ ($C_2\times \GL(2,3)$) x 3, $C_3^2:C_6$ ($C_2^2\times S_4$)

Order 36: $C_6:S_3$ ($C_6\times S_4$) x 3, $C_6:S_3$ ($C_3\times \GL(2,3)$) x 3, $C_6^2$ ($C_3\times \GL(2,3)$)

Order 27: $C_3^3$ ($C_2^2\times \GL(2,3)$)

Order 18: $C_3\times C_6$ ($C_6\times \GL(2,3)$) x 3, $C_3:S_3$ ($C_6\times \GL(2,3)$) x 3, $C_3:S_3$ ($C_2\times C_6\times S_4$)

Order 12: $C_2\times C_6$ ($C_3^2:\GL(2,3)$)

Order 9: $C_3^2$ ($C_2\times C_6\times \GL(2,3)$)

Order 6: $C_6$ ($C_2\times C_3^2:\GL(2,3)$) x 3

Order 4: $C_2^2$ ($C_3^3:\GL(2,3)$)

Order 3: $C_3$ ($C_6^2:\GL(2,3)$)

Order 2: $C_2$ ($C_6\times C_3^2:\GL(2,3)$) x 3

Order 1: $C_1$ ($C_3\times C_6^2:\GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $C_3\times C_6^2:\GL(2,3)$ ($C_1$)

Order 2592: $C_2\times C_6\times \ASL(2,3)$ ($C_2$), $C_6\times C_3^2:\GL(2,3)$ ($C_2$)

Order 1728: $C_6^2:\GL(2,3)$ ($C_3$)

Order 1296: $C_6\times \PU(3,2)$ ($C_2^2$), $C_3^3:\GL(2,3)$ ($C_2^2$)

Order 864: $C_3\times C_6^2:Q_8$ ($S_3$), $C_6^2:\SL(2,3)$ ($C_6$), $C_2\times C_3^2:\GL(2,3)$ ($C_6$)

Order 648: $C_3^3:\SL(2,3)$ ($C_2^3$)

Order 432: $C_6\times \PSU(3,2)$ ($D_6$), $C_2\times \PU(3,2)$ ($C_2\times C_6$), $C_3^2:\GL(2,3)$ ($C_2\times C_6$)

Order 288: $C_6^2:Q_8$ ($C_3\times S_3$)

Order 216: $C_6^2:C_6$ ($S_4$), $C_3^3:Q_8$ ($C_2\times D_6$), $\PU(3,2)$ ($C_2^2\times C_6$)

Order 144: $C_2\times \PSU(3,2)$ ($C_6\times S_3$)

Order 108: $C_3\times C_6^2$ ($\GL(2,3)$), $C_3^2:D_6$ ($C_2\times S_4$), $C_3^2:D_6$ ($\GL(2,3)$)

Order 72: $C_6:D_6$ ($C_3\times S_4$), $\PSU(3,2)$ ($C_6\times D_6$)

Order 54: $C_3^2\times C_6$ ($C_2\times \GL(2,3)$), $C_3^2:C_6$ ($C_2^2\times S_4$), $C_3^2:C_6$ ($C_2\times \GL(2,3)$)

Order 36: $C_6^2$ ($C_3\times \GL(2,3)$), $C_6:S_3$ ($C_6\times S_4$), $C_6:S_3$ ($C_3\times \GL(2,3)$)

Order 27: $C_3^3$ ($C_2^2\times \GL(2,3)$)

Order 18: $C_3\times C_6$ ($C_6\times \GL(2,3)$), $C_3:S_3$ ($C_6\times \GL(2,3)$), $C_3:S_3$ ($C_2\times C_6\times S_4$)

Order 12: $C_2\times C_6$ ($C_3^2:\GL(2,3)$)

Order 9: $C_3^2$ ($C_2\times C_6\times \GL(2,3)$)

Order 6: $C_6$ ($C_2\times C_3^2:\GL(2,3)$)

Order 4: $C_2^2$ ($C_3^3:\GL(2,3)$)

Order 3: $C_3$ ($C_6^2:\GL(2,3)$)

Order 2: $C_2$ ($C_6\times C_3^2:\GL(2,3)$)

Order 1: $C_1$ ($C_3\times C_6^2:\GL(2,3)$)

Series

Derived series

$C_3\times C_6^2:\GL(2,3)$

$\rhd$

$C_3\times C_6^2:\GL(2,3)$

$\rhd$

$\PU(3,2)$

$\rhd$

$\PU(3,2)$

$\rhd$

$\PSU(3,2)$

$\rhd$

$\PSU(3,2)$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3\times C_6^2:\GL(2,3)$

$\rhd$

$C_3\times C_6^2:\GL(2,3)$

$\rhd$

$C_6\times C_3^2:\GL(2,3)$

$\rhd$

$C_6\times C_3^2:\GL(2,3)$

$\rhd$

$C_6\times \PU(3,2)$

$\rhd$

$C_6\times \PU(3,2)$

$\rhd$

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

$\rhd$

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

$\rhd$

$\PU(3,2)$

$\rhd$

$\PU(3,2)$

$\rhd$

$\PSU(3,2)$

$\rhd$

$\PSU(3,2)$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3\times C_6^2:\GL(2,3)$

$\rhd$

$C_3\times C_6^2:\GL(2,3)$

$\rhd$

$\PU(3,2)$

$\rhd$

$\PU(3,2)$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2\times C_6$

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

See the $132 \times 132$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $80 \times 80$ rational character table.