Abstract group 23328.jz: $C_3^4:C_{24}:D_6$

• Label: 23328.jz
• Order: \( 2^{5} \cdot 3^{6} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $81$
• Trans deg.: $81$
• Rank: $3$

Group information

Description:	$C_3^4:C_{24}:D_6$
Order:	\(23328\)\(\medspace = 2^{5} \cdot 3^{6} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^4.C_4:S_3^2.C_2^2$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
Composition factors:	$C_2$ x 5, $C_3$ x 6
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	459	728	3564	7560	3240	5184	2592	23328
Conjugacy classes	1	5	8	4	13	4	6	8	49
Divisions	1	5	8	4	13	2	5	3	41
Autjugacy classes	1	4	6	3	8	4	3	4	33

Dimension	1	2	4	8	16	32	72
Irr. complex chars.	8	14	14	4	4	1	4	49
Irr. rational chars.	8	10	6	7	5	1	4	41

Minimal presentations

Permutation degree:	$81$
Transitive degree:	$81$
Rank:	$3$
Inequivalent generating triples:	$69672960$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	72	72	72
Arbitrary	72	72	72

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{24}=d^{3}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $81$ $\langle(2,9)(4,17)(5,8)(6,21)(7,28)(10,33)(13,40)(15,48)(16,26)(18,32)(19,50)(20,55) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4:C_{24})$ $\,\rtimes\,$ $D_6$ (2)	$C_3^4$ $\,\rtimes\,$ $(C_{24}:D_6)$	$(C_3^2:F_9)$ $\,\rtimes\,$ $S_3^2$	$C_3^4$ $\,\rtimes\,$ $(F_9:C_2^2)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $(F_9:D_6)$ (2)	$(C_3^4:D_{12})$ . $D_6$ (2)	$C_3^2$ . $(F_9:S_3^2)$	$(C_3^4:C_3.D_6)$ . $D_4$ (2)	all 12
Aut. group:	$\Aut(C_3^4:C_{12})$	$\Aut(C_3^4:C_{24})$	$\Aut(C_3^4:D_{12})$	$\Aut(C_3^3:\PSU(3,2))$	all 5

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 115140 subgroups in 899 conjugacy classes, 46 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^4:C_{24}:D_6$
Commutator:	$G' \simeq$ $C_3^4:C_3:C_2.C_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $F_9:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4:C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2\times \SD_{16}$
Radical:	$R \simeq$ $C_3^4:C_{24}:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $F_9:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3^2$

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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Classes of subgroups up to conjugation

Order 23328: $C_3^4:C_{24}:D_6$

Order 11664: $C_3^4:C_3.C_6.D_4$ x 2, $C_3^4.(C_2\times F_9)$, $C_3^4:C_3.C_4.C_6.C_2$, $C_3^4.C_{12}.C_6.C_2$, $C_3^4.C_3:S_3.C_2^3$, $C_3^3.S_3^3.C_2$

Order 7776: $C_3^4:(S_3\times \SD_{16})$ x 2

Order 5832: $C_3^4:C_3.C_6.C_2^2$ x 2, $C_3^4:C_3.C_{12}.C_2$ x 2, $C_3^4.C_3^2.Q_8$, $C_3^3.S_3^3$, $C_3^4.C_3:S_3.C_2^2$, $C_3^4.C_{12}.C_6$, $C_3^4.C_3^2.C_8$, $C_3^4.S_3\wr C_2$, $C_3^4.C_3:S_3.C_2^2$

Order 3888: $C_3^3:(C_2\times F_9)$ x 3, $C_3^2:F_9:S_3$ x 2, $C_3^3:F_9:C_2$ x 2, $C_3^4:(C_3\times \SD_{16})$ x 2, $C_3^4:(S_3\times Q_8)$ x 2, $C_3^3:F_9:C_2$ x 2, $C_3^4:(S_3\times D_4)$ x 2

Order 2916: $C_3^4.(C_6\times S_3)$ x 2, $C_3^4:C_3.C_{12}$ x 2, $C_3^4:C_3.C_6.C_2$, $C_3^3.C_3^3.C_4$, $C_3^4:C_3.D_6$, $C_3^4.S_3^2$, $C_3^4:C_3:C_2.C_6$

Order 2592: $F_9:S_3^2$, $C_3^4:(C_2\times \SD_{16})$

Order 1944: $C_3^4:(C_4\times S_3)$ x 5, $C_3^3:\PSU(3,2)$ x 4, $C_3^2.S_3^3$ x 3, $C_3^3:F_9$ x 3, $C_3^4:C_{24}$ x 3, $C_3^3:\PSU(3,2)$ x 2, $C_3^4:D_{12}$ x 2, $C_3^3:\SOPlus(4,2)$ x 2, $C_3^3:\SOPlus(4,2)$ x 2

Order 1458: $C_3^4:(C_3\times C_6)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:C_3:S_3$

Order 1296: $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:(C_2\times Q_8)$, $C_3^3:(S_3\times Q_8)$, $C_3^4:(C_2\times D_4)$, $C_3^3:(S_3\times D_4)$, $C_3^2:(C_2\times F_9)$, $C_3:S_3\times F_9$

Order 972: $C_3^3:S_3^2$ x 6, $C_3^4:C_3:C_4$ x 5, $C_3^4:C_{12}$ x 5, $C_3^4:D_6$ x 4, $C_3^4:D_6$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3

Order 864: $F_9:D_6$ x 2

Order 729: $C_3^4:C_3^2$

Order 648: $C_3^4:Q_8$ x 3, $C_3^4:D_4$ x 3, $C_3:S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:(C_2\times C_4)$ x 2, $C_3.S_3^3$ x 2, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^2\times F_9$

Order 486: $C_3^4:S_3$ x 7, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 4, $C_3^4:S_3$ x 4

Order 432: $S_3\times F_9$ x 3, $S_3\times \PSU(3,2)$ x 2, $S_3^3:C_2$ x 2, $F_9:S_3$ x 2, $C_3^3:\SD_{16}$ x 2, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2

Order 324: $C_3^3:D_6$ x 7, $C_3\wr C_2^2$ x 6, $C_3^2:S_3^2$ x 5, $C_3^4:C_4$ x 4, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3\wr C_4$ x 2, $C_3^2:S_3^2$ x 2, $C_3^4:C_4$, $C_3^3:C_{12}$

Order 288: $F_9:C_2^2$ x 2, $C_{24}:D_6$

Order 243: $C_3^4:C_3$ x 7

Order 216: $S_3^3$ x 5, $C_3^2:C_4\times S_3$ x 5, $C_3^3:Q_8$ x 4, $C_3:F_9$ x 3, $C_3\times F_9$ x 3, $C_3^3:Q_8$ x 2, $C_3^2:D_{12}$ x 2, $S_3^2:S_3$ x 2, $S_3^2:C_6$ x 2, $C_6.S_3^2$ x 2

Order 162: $C_3^3:C_6$ x 14, $C_3^3:S_3$ x 12, $C_3^2\wr C_2$ x 11, $C_3^3:C_6$ x 6, $C_3^3:S_3$ x 2, $C_3^3:S_3$ x 2, $S_3\times \He_3$ x 2

Order 144: $F_9:C_2$ x 8, $C_{24}:C_6$ x 2, $C_{12}.D_6$ x 2, $C_2\times \PSU(3,2)$ x 2, $S_3^2:C_2^2$ x 2, $C_2\times F_9$ x 2, $C_{24}:S_3$, $C_{12}:D_6$, $C_{12}.D_6$

Order 108: $C_3\times S_3^2$ x 16, $C_3:S_3^2$ x 14, $C_3^2:D_6$ x 14, $C_3:S_3^2$ x 6, $C_3^3:C_4$ x 5, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 4, $C_3^2:D_6$ x 2

Order 96: $S_3\times \SD_{16}$ x 2

Order 81: $C_3\times \He_3$ x 12, $C_3^4$ x 3

Order 72: $S_3\times D_6$ x 7, $\PSU(3,2)$ x 7, $\SOPlus(4,2)$ x 7, $C_2\times C_3^2:C_4$ x 4, $F_9$ x 4, $C_3\times D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_{12}:S_3$, $C_3^2:Q_8$, $D_6:S_3$, $C_6.D_6$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 54: $C_3^2:C_6$ x 21, $S_3\times C_3^2$ x 21, $C_3^2:C_6$ x 20, $C_3^2:S_3$ x 14, $C_3^2:S_3$ x 6, $C_2\times \He_3$ x 2

Order 48: $S_3\times C_8$ x 3, $C_{24}:C_2$ x 2, $S_3\times Q_8$ x 2, $S_3\times D_4$ x 2, $C_3\times \SD_{16}$ x 2, $Q_8:S_3$ x 2, $C_3:\SD_{16}$ x 2

Order 36: $S_3^2$ x 32, $C_6\times S_3$ x 14, $C_3^2:C_4$ x 12, $C_6:S_3$ x 7, $C_3:C_{12}$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2\times \SD_{16}$

Order 27: $C_3^3$ x 12, $\He_3$ x 12

Order 24: $C_4\times S_3$ x 5, $C_3:Q_8$ x 4, $C_2\times D_6$ x 4, $C_{24}$ x 3, $C_3:C_8$ x 3, $C_3:D_4$ x 2, $D_{12}$ x 2, $C_3\times Q_8$ x 2, $C_3\times D_4$ x 2

Order 18: $C_3\times S_3$ x 48, $C_3:S_3$ x 22, $C_3\times C_6$ x 9

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

Order 12: $D_6$ x 23, $C_{12}$ x 5, $C_3:C_4$ x 5, $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 26

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

Order 6: $S_3$ x 21, $C_6$ x 13

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 3: $C_3$ x 8

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 23328: $C_3^4:C_{24}:D_6$

Order 11664: $C_3^4.(C_2\times F_9)$, $C_3^4.C_{12}.C_6.C_2$, $C_3^4:C_3.C_6.D_4$, $C_3^4.C_3:S_3.C_2^3$, $C_3^3.S_3^3.C_2$

Order 7776: $C_3^4:(S_3\times \SD_{16})$

Order 5832: $C_3^4.C_3^2.Q_8$, $C_3^4:C_3.C_6.C_2^2$, $C_3^3.S_3^3$, $C_3^4.C_3:S_3.C_2^2$, $C_3^4.C_{12}.C_6$, $C_3^4.C_3^2.C_8$, $C_3^4:C_3.C_{12}.C_2$, $C_3^4.S_3\wr C_2$, $C_3^4.C_3:S_3.C_2^2$

Order 3888: $C_3^3:(C_2\times F_9)$ x 2, $C_3^2:F_9:S_3$, $C_3^3:F_9:C_2$, $C_3^4:(C_3\times \SD_{16})$, $C_3^4:(S_3\times Q_8)$, $C_3^3:F_9:C_2$, $C_3^4:(S_3\times D_4)$

Order 2916: $C_3^4:C_3.C_6.C_2$, $C_3^4.(C_6\times S_3)$, $C_3^3.C_3^3.C_4$, $C_3^4:C_3.D_6$, $C_3^4:C_3.C_{12}$, $C_3^4.S_3^2$, $C_3^4:C_3:C_2.C_6$

Order 2592: $F_9:S_3^2$, $C_3^4:(C_2\times \SD_{16})$

Order 1944: $C_3^4:(C_4\times S_3)$ x 3, $C_3^2.S_3^3$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^3:F_9$ x 2, $C_3^4:C_{24}$ x 2, $C_3^3:\PSU(3,2)$, $C_3^4:D_{12}$, $C_3^3:\SOPlus(4,2)$, $C_3^3:\SOPlus(4,2)$

Order 1458: $C_3^4:(C_3\times C_6)$ x 2, $C_3^4:C_3:S_3$, $C_3^4:(C_3\times S_3)$

Order 1296: $C_3^4:(C_2\times Q_8)$, $C_3^3:(S_3\times Q_8)$, $C_3^4:(C_2\times D_4)$, $C_3^3:(S_3\times D_4)$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^2:(C_2\times F_9)$, $C_3:S_3\times F_9$

Order 972: $C_3^4:D_6$ x 3, $C_3^3:S_3^2$ x 3, $C_3^4:C_3:C_4$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$ x 2

Order 864: $F_9:D_6$

Order 729: $C_3^4:C_3^2$

Order 648: $C_3:S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^4:D_4$ x 2, $C_3^4:(C_2\times C_4)$ x 2, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^2\times F_9$, $C_3.S_3^3$

Order 486: $C_3^4:S_3$ x 5, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 3, $C_3^4:S_3$ x 3, $C_3^4:C_6$ x 3

Order 432: $S_3\times F_9$ x 2, $S_3\times \PSU(3,2)$, $S_3^3:C_2$, $F_9:S_3$, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$, $F_9:C_6$

Order 324: $C_3\wr C_2^2$ x 5, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$ x 4, $C_3^4:C_4$ x 3, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^4:C_4$, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^2:S_3^2$

Order 288: $F_9:C_2^2$ x 2, $C_{24}:D_6$

Order 243: $C_3^4:C_3$ x 5

Order 216: $S_3^3$ x 4, $C_3^2:C_4\times S_3$ x 3, $C_3^3:Q_8$ x 2, $C_3:F_9$ x 2, $C_3\times F_9$ x 2, $C_3^3:Q_8$, $C_3^2:D_{12}$, $S_3^2:S_3$, $S_3^2:C_6$, $C_6.S_3^2$

Order 162: $C_3^2\wr C_2$ x 8, $C_3^3:S_3$ x 8, $C_3^3:C_6$ x 7, $C_3^3:C_6$ x 4, $C_3^3:S_3$ x 2, $C_3^3:S_3$, $S_3\times \He_3$

Order 144: $F_9:C_2$ x 4, $C_2\times \PSU(3,2)$ x 2, $S_3^2:C_2^2$ x 2, $C_2\times F_9$ x 2, $C_{24}:S_3$, $C_{24}:C_6$, $C_{12}.D_6$, $C_{12}:D_6$, $C_{12}.D_6$

Order 108: $C_3\times S_3^2$ x 10, $C_3:S_3^2$ x 9, $C_3^2:D_6$ x 8, $C_3:S_3^2$ x 5, $C_3^3:C_4$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3^2:D_6$

Order 96: $S_3\times \SD_{16}$

Order 81: $C_3\times \He_3$ x 8, $C_3^4$ x 3

Order 72: $S_3\times D_6$ x 5, $\PSU(3,2)$ x 5, $\SOPlus(4,2)$ x 5, $C_2\times C_3^2:C_4$ x 4, $F_9$ x 4, $C_{12}:S_3$, $C_3\times D_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $D_6:S_3$, $C_6.D_6$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 54: $C_3^2:C_6$ x 14, $S_3\times C_3^2$ x 13, $C_3^2:C_6$ x 11, $C_3^2:S_3$ x 9, $C_3^2:S_3$ x 5, $C_2\times \He_3$

Order 48: $S_3\times C_8$ x 2, $C_{24}:C_2$, $S_3\times Q_8$, $S_3\times D_4$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_3:\SD_{16}$

Order 36: $S_3^2$ x 20, $C_3^2:C_4$ x 9, $C_6\times S_3$ x 8, $C_6:S_3$ x 5, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2\times \SD_{16}$

Order 27: $C_3^3$ x 10, $\He_3$ x 8

Order 24: $C_4\times S_3$ x 3, $C_2\times D_6$ x 3, $C_3:Q_8$ x 2, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_3:D_4$, $D_{12}$, $C_3\times Q_8$, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 27, $C_3:S_3$ x 17, $C_3\times C_6$ x 6

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

Order 12: $D_6$ x 14, $C_2\times C_6$ x 3, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 19

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

Order 6: $S_3$ x 14, $C_6$ x 8

Order 4: $C_2^2$ x 4, $C_4$ x 3

Order 3: $C_3$ x 6

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 23328: $C_3^4:C_{24}:D_6$ ($C_1$)

Order 11664: $C_3^4:C_3.C_6.D_4$ ($C_2$) x 2, $C_3^4.(C_2\times F_9)$ ($C_2$), $C_3^4:C_3.C_4.C_6.C_2$ ($C_2$), $C_3^4.C_{12}.C_6.C_2$ ($C_2$), $C_3^4.C_3:S_3.C_2^3$ ($C_2$), $C_3^3.S_3^3.C_2$ ($C_2$)

Order 5832: $C_3^4:C_3.C_6.C_2^2$ ($C_2^2$) x 2, $C_3^4:C_3.C_{12}.C_2$ ($C_2^2$) x 2, $C_3^4.C_{12}.C_6$ ($C_2^2$), $C_3^4.C_3^2.C_8$ ($C_2^2$), $C_3^4.C_3:S_3.C_2^2$ ($C_2^2$)

Order 3888: $C_3^2:F_9:S_3$ ($S_3$) x 2

Order 2916: $C_3^3.C_3^3.C_4$ ($D_4$), $C_3^4:C_3.D_6$ ($D_4$), $C_3^4:C_3:C_2.C_6$ ($C_2^3$)

Order 1944: $C_3^3:\PSU(3,2)$ ($D_6$) x 2, $C_3^4:D_{12}$ ($D_6$) x 2, $C_3^4:C_{24}$ ($D_6$) x 2

Order 1458: $C_3^4:C_3:S_3$ ($\SD_{16}$), $C_3^4:(C_3\times C_6)$ ($\SD_{16}$), $C_3^4:(C_3\times C_6)$ ($C_2\times D_4$)

Order 972: $C_3^4:C_{12}$ ($C_2\times D_6$) x 2

Order 729: $C_3^4:C_3^2$ ($C_2\times \SD_{16}$)

Order 648: $C_3^2:F_9$ ($S_3^2$)

Order 486: $C_3^4:C_6$ ($S_3\times D_4$) x 2

Order 324: $C_3^4:C_4$ ($S_3\times D_6$)

Order 243: $C_3^4:C_3$ ($S_3\times \SD_{16}$) x 2

Order 162: $C_3^3:S_3$ ($F_9:C_2$), $C_3^3:S_3$ ($C_{12}:D_6$)

Order 81: $C_3^4$ ($C_{24}:D_6$), $C_3^4$ ($F_9:C_2^2$)

Order 27: $C_3^3$ ($F_9:D_6$) x 2

Order 9: $C_3^2$ ($F_9:S_3^2$)

Order 1: $C_1$ ($C_3^4:C_{24}:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 23328: $C_3^4:C_{24}:D_6$ ($C_1$)

Order 11664: $C_3^4.(C_2\times F_9)$ ($C_2$), $C_3^4.C_{12}.C_6.C_2$ ($C_2$), $C_3^4:C_3.C_6.D_4$ ($C_2$), $C_3^4.C_3:S_3.C_2^3$ ($C_2$), $C_3^3.S_3^3.C_2$ ($C_2$)

Order 5832: $C_3^4:C_3.C_6.C_2^2$ ($C_2^2$), $C_3^4.C_{12}.C_6$ ($C_2^2$), $C_3^4.C_3^2.C_8$ ($C_2^2$), $C_3^4:C_3.C_{12}.C_2$ ($C_2^2$), $C_3^4.C_3:S_3.C_2^2$ ($C_2^2$)

Order 3888: $C_3^2:F_9:S_3$ ($S_3$)

Order 2916: $C_3^3.C_3^3.C_4$ ($D_4$), $C_3^4:C_3.D_6$ ($D_4$), $C_3^4:C_3:C_2.C_6$ ($C_2^3$)

Order 1944: $C_3^3:\PSU(3,2)$ ($D_6$), $C_3^4:D_{12}$ ($D_6$), $C_3^4:C_{24}$ ($D_6$)

Order 1458: $C_3^4:C_3:S_3$ ($\SD_{16}$), $C_3^4:(C_3\times C_6)$ ($\SD_{16}$), $C_3^4:(C_3\times C_6)$ ($C_2\times D_4$)

Order 972: $C_3^4:C_{12}$ ($C_2\times D_6$)

Order 729: $C_3^4:C_3^2$ ($C_2\times \SD_{16}$)

Order 648: $C_3^2:F_9$ ($S_3^2$)

Order 486: $C_3^4:C_6$ ($S_3\times D_4$)

Order 324: $C_3^4:C_4$ ($S_3\times D_6$)

Order 243: $C_3^4:C_3$ ($S_3\times \SD_{16}$)

Order 162: $C_3^3:S_3$ ($F_9:C_2$), $C_3^3:S_3$ ($C_{12}:D_6$)

Order 81: $C_3^4$ ($C_{24}:D_6$), $C_3^4$ ($F_9:C_2^2$)

Order 27: $C_3^3$ ($F_9:D_6$)

Order 9: $C_3^2$ ($F_9:S_3^2$)

Order 1: $C_1$ ($C_3^4:C_{24}:D_6$)

Series

Derived series

$C_3^4:C_{24}:D_6$

$\rhd$

$C_3^4:C_3:C_2.C_6$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^4:C_{24}:D_6$

$\rhd$

$C_3^4.(C_2\times F_9)$

$\rhd$

$C_3^4.C_{12}.C_6$

$\rhd$

$C_3^4:C_{24}$

$\rhd$

$C_3^4:C_{12}$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_3^4:C_{24}:D_6$

$\rhd$

$C_3^4:C_3:C_2.C_6$

$\rhd$

$C_3^4:(C_3\times C_6)$

$\rhd$

$C_3^4:C_3^2$

Upper central series

$C_1$

Character theory

Complex character table

See the $49 \times 49$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $41 \times 41$ rational character table.