Abstract group 1728.46792: $C_6\times D_6:S_4$

• Label: 1728.46792
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $16$
• Trans deg.: $144$
• Rank: $3$

Group information

Description:	$C_6\times D_6:S_4$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_6\times A_4).C_2^6.C_2$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	135	80	120	864	528	1728
Conjugacy classes	1	13	11	6	77	36	144
Divisions	1	13	7	6	45	14	86
Autjugacy classes	1	8	7	3	28	7	54

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	24	54	24	12	30	0	0	0	144
Irr. rational chars.	8	18	8	22	14	6	8	2	86

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$144$
Rank:	$3$
Inequivalent generating triples:	$131040$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	7	9

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{12}=c^{6}=d^{2}=e^{6}=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(9,10)(11,12)(14,16), (2,4)(13,14,15,16), (11,12), (5,6,7), (13,15)(14,16), (2,3,4), (8,9,10), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 2 & 21 \\ 21 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 7 & 2 \\ 18 & 21 \end{array}\right), \left(\begin{array}{rr} 21 & 2 \\ 18 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $(D_6:S_4)$
Semidirect product:	$(C_6\times D_6)$ $\,\rtimes\,$ $S_4$	$D_6$ $\,\rtimes\,$ $(C_6\times S_4)$ (2)	$(A_4:C_{12})$ $\,\rtimes\,$ $D_6$ (2)	$(C_6\times A_4)$ $\,\rtimes\,$ $D_{12}$ (2)	all 44
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(D_6\times S_4)$	$C_6^2$ . $(C_2\times S_4)$	$C_2$ . $(C_6\times S_3\times S_4)$	$C_6$ . $(C_2\times C_6\times S_4)$	all 24

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{28}\Z)$.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 10196 subgroups in 1295 conjugacy classes, 106 normal (50 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $S_3\times S_4$
Commutator:	$G' \simeq$ $C_6\times A_4$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6\times S_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6\times D_6:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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 1728: $C_6\times D_6:S_4$

Order 864: $C_3^2:\GL(2,\mathbb{Z}/4)$ x 4, $C_6\times A_4\times D_6$, $C_6^2:S_4$, $C_6^2.S_4$

Order 576: $C_6\times \GL(2,\mathbb{Z}/4)$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_6^2.C_2^4$

Order 432: $C_6\times S_3\times A_4$ x 4, $C_6^2:D_6$ x 4, $C_3\times A_4:C_{12}$ x 2, $A_4\times C_6^2$, $C_6^2.D_6$

Order 288: $C_6^2:D_4$ x 8, $C_3\times \GL(2,\mathbb{Z}/4)$ x 4, $D_6:S_4$ x 4, $C_2\times A_4:C_{12}$ x 3, $C_6^2:D_4$ x 2, $C_6^2.D_4$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6\times S_4$ x 2, $C_6^2.D_4$, $C_6^2:C_2^3$, $C_2^3:C_6^2$, $C_2\times C_6:S_4$, $C_6^2:D_4$

Order 216: $C_3^2:D_{12}$ x 4, $C_6^2:C_6$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6.C_6^2$

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^5:C_6$, $C_2^3:D_{12}$

Order 144: $C_6\times D_{12}$ x 16, $C_6^2:C_2^2$ x 11, $C_2^2:C_6^2$ x 10, $A_4\times D_6$ x 10, $C_6\times S_4$ x 10, $C_6^2:C_2^2$ x 9, $D_6:C_{12}$ x 8, $A_4:C_{12}$ x 6, $C_6:S_4$ x 4, $C_6^2:C_4$ x 4, $C_2\times C_6\times C_{12}$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_4$, $C_6:D_{12}$

Order 108: $C_3^2:D_6$ x 4, $C_3^2\times D_6$ x 4, $C_3^2:C_{12}$ x 2, $C_3\times C_6^2$, $C_3^2\times A_4$

Order 96: $C_2^2:D_{12}$ x 8, $C_2^4:C_6$ x 8, $C_2^3:C_{12}$ x 5, $\GL(2,\mathbb{Z}/4)$ x 4, $C_{12}:C_2^3$ x 3, $C_2^3\times A_4$ x 2, $C_2^2\times S_4$ x 2, $C_2^2\times D_{12}$ x 2, $C_2^2.D_{12}$ x 2, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_2^3:D_6$, $C_2^2.S_4$

Order 72: $C_6\times D_6$ x 43, $C_6\times A_4$ x 20, $C_3\times D_{12}$ x 16, $C_6\wr C_2$ x 12, $C_2\times C_6^2$ x 8, $C_6\times C_{12}$ x 8, $C_6:C_{12}$ x 7, $S_3\times A_4$ x 6, $C_3\times S_4$ x 6, $C_3:D_{12}$ x 4, $C_3:S_4$ x 2, $C_6:D_6$

Order 64: $C_2^3:D_4$

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

Order 48: $C_6\times D_4$ x 24, $C_2^2:C_{12}$ x 20, $C_2\times D_{12}$ x 16, $C_2^3\times C_6$ x 14, $C_2^2\times A_4$ x 14, $C_2^2\times D_6$ x 11, $C_2\times S_4$ x 10, $C_6:D_4$ x 9, $D_6:C_4$ x 8, $C_2^2\times C_{12}$ x 7, $A_4:C_4$ x 2, $C_6.C_2^3$

Order 36: $C_6\times S_3$ x 56, $C_6^2$ x 22, $C_3:C_{12}$ x 8, $C_3\times A_4$ x 6, $C_3\times C_{12}$ x 4, $C_6:S_3$ x 4

Order 32: $C_2^2\wr C_2$ x 8, $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 62, $C_2\times D_6$ x 41, $C_2\times C_{12}$ x 28, $C_3\times D_4$ x 24, $C_2\times A_4$ x 18, $D_{12}$ x 16, $C_3:D_4$ x 12, $S_4$ x 6, $C_6:C_4$ x 5

Order 18: $C_3\times C_6$ x 27, $C_3\times S_3$ x 18, $C_3:S_3$ x 2

Order 16: $C_2\times D_4$ x 24, $C_2^2:C_4$ x 12, $C_2^4$ x 12, $C_2^2\times C_4$ x 3

Order 12: $C_2\times C_6$ x 86, $D_6$ x 46, $C_{12}$ x 14, $A_4$ x 4, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 46, $D_4$ x 24, $C_2\times C_4$ x 12

Order 6: $C_6$ x 45, $S_3$ x 12

Order 4: $C_2^2$ x 48, $C_4$ x 6

Order 3: $C_3$ x 7

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_6\times D_6:S_4$

Order 864: $C_6\times A_4\times D_6$, $C_6^2:S_4$, $C_6^2.S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$

Order 576: $C_6\times \GL(2,\mathbb{Z}/4)$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_6^2.C_2^4$

Order 432: $C_6\times S_3\times A_4$ x 2, $C_6^2:D_6$ x 2, $A_4\times C_6^2$, $C_6^2.D_6$, $C_3\times A_4:C_{12}$

Order 288: $C_2\times A_4:C_{12}$ x 3, $C_6^2:D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2:D_4$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6\times S_4$ x 2, $C_3\times \GL(2,\mathbb{Z}/4)$, $D_6:S_4$, $C_6^2.D_4$, $C_6^2:C_2^3$, $C_2^3:C_6^2$, $C_2\times C_6:S_4$, $C_6^2:D_4$

Order 216: $C_6^2:C_6$ x 2, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6^2:C_6$, $C_3^2:S_4$, $C_6.C_6^2$, $C_3^2:D_{12}$

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^5:C_6$, $C_2^3:D_{12}$

Order 144: $C_2^2:C_6^2$ x 8, $C_6^2:C_2^2$ x 7, $C_6\times D_{12}$ x 6, $C_6^2:C_2^2$ x 5, $A_4\times D_6$ x 4, $C_6\times S_4$ x 4, $A_4:C_{12}$ x 3, $D_6:C_{12}$ x 2, $C_6:S_4$ x 2, $C_2\times C_6\times C_{12}$ x 2, $C_6^2:C_4$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_4$, $C_6:D_{12}$

Order 108: $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3\times C_6^2$, $C_3^2\times A_4$, $C_3^2:C_{12}$

Order 96: $C_2^3:C_{12}$ x 5, $C_{12}:C_2^3$ x 3, $C_2^2:D_{12}$ x 2, $C_2^3\times A_4$ x 2, $C_2^2\times S_4$ x 2, $C_2^2\times D_{12}$ x 2, $C_2^4:C_6$ x 2, $C_2^2.D_{12}$ x 2, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_2^3:D_6$, $\GL(2,\mathbb{Z}/4)$, $C_2^2.S_4$

Order 72: $C_6\times D_6$ x 23, $C_6\times A_4$ x 13, $C_2\times C_6^2$ x 6, $C_6\times C_{12}$ x 5, $C_6:C_{12}$ x 5, $C_6\wr C_2$ x 4, $C_3\times D_{12}$ x 3, $S_3\times A_4$ x 2, $C_3\times S_4$ x 2, $C_6:D_6$, $C_3:S_4$, $C_3:D_{12}$

Order 64: $C_2^3:D_4$

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

Order 48: $C_2^3\times C_6$ x 10, $C_6\times D_4$ x 10, $C_2^2\times A_4$ x 8, $C_2^2:C_{12}$ x 8, $C_2^2\times D_6$ x 7, $C_2^2\times C_{12}$ x 7, $C_2\times D_{12}$ x 6, $C_6:D_4$ x 5, $C_2\times S_4$ x 4, $D_6:C_4$ x 2, $C_6.C_2^3$, $A_4:C_4$

Order 36: $C_6\times S_3$ x 23, $C_6^2$ x 17, $C_3\times A_4$ x 6, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 2, $C_6:S_3$ x 2

Order 32: $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^2\wr C_2$ x 2, $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 36, $C_2\times D_6$ x 21, $C_2\times C_{12}$ x 17, $C_2\times A_4$ x 10, $C_3\times D_4$ x 6, $C_3:D_4$ x 4, $C_6:C_4$ x 3, $D_{12}$ x 3, $S_4$ x 2

Order 18: $C_3\times C_6$ x 18, $C_3\times S_3$ x 7, $C_3:S_3$

Order 16: $C_2\times D_4$ x 10, $C_2^4$ x 8, $C_2^2:C_4$ x 4, $C_2^2\times C_4$ x 3

Order 12: $C_2\times C_6$ x 50, $D_6$ x 19, $C_{12}$ x 7, $A_4$ x 4, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 24, $C_2\times C_4$ x 7, $D_4$ x 6

Order 6: $C_6$ x 28, $S_3$ x 5

Order 4: $C_2^2$ x 24, $C_4$ x 3

Order 3: $C_3$ x 7

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_6\times D_6:S_4$ ($C_1$)

Order 864: $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$) x 4, $C_6\times A_4\times D_6$ ($C_2$), $C_6^2:S_4$ ($C_2$), $C_6^2.S_4$ ($C_2$)

Order 576: $C_6:\GL(2,\mathbb{Z}/4)$ ($C_3$)

Order 432: $C_6\times S_3\times A_4$ ($C_2^2$) x 2, $C_6^2:D_6$ ($C_2^2$) x 2, $C_3\times A_4:C_{12}$ ($C_2^2$) x 2, $A_4\times C_6^2$ ($C_2^2$)

Order 288: $D_6:S_4$ ($C_6$) x 4, $C_2\times A_4:C_{12}$ ($C_6$), $C_2\times A_4:C_{12}$ ($S_3$), $C_6^2:C_2^3$ ($S_3$), $C_2\times A_4\times D_6$ ($C_6$), $C_2\times C_6:S_4$ ($C_6$)

Order 216: $C_6^2:C_6$ ($D_4$) x 2, $C_6^2:C_6$ ($C_2^3$)

Order 144: $C_6^2:C_2^2$ ($D_6$) x 2, $A_4\times D_6$ ($C_2\times C_6$) x 2, $C_6:S_4$ ($C_2\times C_6$) x 2, $A_4:C_{12}$ ($C_2\times C_6$) x 2, $A_4:C_{12}$ ($D_6$) x 2, $C_2^2\times C_6^2$ ($D_6$), $C_2^2:C_6^2$ ($C_2\times C_6$), $C_2^2:C_6^2$ ($D_6$)

Order 108: $C_3^2\times A_4$ ($C_2\times D_4$)

Order 96: $C_2^3\times D_6$ ($C_3\times S_3$), $C_2^2.S_4$ ($C_3\times S_3$)

Order 72: $C_2\times C_6^2$ ($C_3:D_4$) x 2, $C_6\times A_4$ ($D_{12}$) x 2, $C_6\times A_4$ ($C_3\times D_4$) x 2, $C_2\times C_6^2$ ($C_2\times D_6$), $C_6\times D_6$ ($S_4$), $C_6\times A_4$ ($C_2^2\times C_6$), $C_6\times A_4$ ($C_2\times D_6$)

Order 48: $C_2^2\times D_6$ ($C_6\times S_3$) x 2, $A_4:C_4$ ($C_6\times S_3$) x 2, $C_2^3\times C_6$ ($C_6\times S_3$), $C_2^3\times C_6$ ($S_3^2$), $C_2^2\times A_4$ ($C_6\times S_3$)

Order 36: $C_6\times S_3$ ($C_2\times S_4$) x 2, $C_6^2$ ($C_2\times S_4$), $C_6^2$ ($C_6:D_4$), $C_3\times A_4$ ($C_6\times D_4$), $C_3\times A_4$ ($C_2\times D_{12}$)

Order 24: $C_2^2\times C_6$ ($C_6\wr C_2$) x 2, $C_2^2\times C_6$ ($C_3:D_{12}$) x 2, $C_2\times A_4$ ($C_3\times D_{12}$) x 2, $C_2^2\times C_6$ ($C_6\times D_6$), $C_2^2\times C_6$ ($S_3\times D_6$), $C_2\times D_6$ ($C_3\times S_4$), $C_2\times A_4$ ($C_6\times D_6$)

Order 18: $C_3\times C_6$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_3\times C_6$ ($C_2^2\times S_4$)

Order 16: $C_2^4$ ($C_3\times S_3^2$)

Order 12: $D_6$ ($C_6\times S_4$) x 2, $C_2\times C_6$ ($C_6\times S_4$), $C_2\times C_6$ ($S_3\times S_4$), $C_2\times C_6$ ($C_6^2:C_2^2$), $C_2\times C_6$ ($C_6:D_{12}$), $A_4$ ($C_6\times D_{12}$)

Order 9: $C_3^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_3^2:D_{12}$) x 2, $C_2^3$ ($C_6\times S_3^2$)

Order 6: $C_6$ ($C_3\times \GL(2,\mathbb{Z}/4)$) x 2, $C_6$ ($D_6:S_4$) x 2, $C_6$ ($C_2\times C_6\times S_4$), $C_6$ ($D_6\times S_4$)

Order 4: $C_2^2$ ($C_6^2:D_6$), $C_2^2$ ($C_6^2.D_6$)

Order 3: $C_3$ ($C_6\times \GL(2,\mathbb{Z}/4)$), $C_3$ ($C_6:\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$) x 2, $C_2$ ($C_6\times S_3\times S_4$)

Order 1: $C_1$ ($C_6\times D_6:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_6\times D_6:S_4$ ($C_1$)

Order 864: $C_6\times A_4\times D_6$ ($C_2$), $C_6^2:S_4$ ($C_2$), $C_6^2.S_4$ ($C_2$), $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$)

Order 576: $C_6:\GL(2,\mathbb{Z}/4)$ ($C_3$)

Order 432: $A_4\times C_6^2$ ($C_2^2$), $C_6\times S_3\times A_4$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $C_3\times A_4:C_{12}$ ($C_2^2$)

Order 288: $C_2\times A_4:C_{12}$ ($C_6$), $C_2\times A_4:C_{12}$ ($S_3$), $D_6:S_4$ ($C_6$), $C_6^2:C_2^3$ ($S_3$), $C_2\times A_4\times D_6$ ($C_6$), $C_2\times C_6:S_4$ ($C_6$)

Order 216: $C_6^2:C_6$ ($C_2^3$), $C_6^2:C_6$ ($D_4$)

Order 144: $C_2^2\times C_6^2$ ($D_6$), $C_6^2:C_2^2$ ($D_6$), $C_2^2:C_6^2$ ($C_2\times C_6$), $C_2^2:C_6^2$ ($D_6$), $A_4\times D_6$ ($C_2\times C_6$), $C_6:S_4$ ($C_2\times C_6$), $A_4:C_{12}$ ($C_2\times C_6$), $A_4:C_{12}$ ($D_6$)

Order 108: $C_3^2\times A_4$ ($C_2\times D_4$)

Order 96: $C_2^3\times D_6$ ($C_3\times S_3$), $C_2^2.S_4$ ($C_3\times S_3$)

Order 72: $C_2\times C_6^2$ ($C_3:D_4$), $C_2\times C_6^2$ ($C_2\times D_6$), $C_6\times D_6$ ($S_4$), $C_6\times A_4$ ($D_{12}$), $C_6\times A_4$ ($C_2^2\times C_6$), $C_6\times A_4$ ($C_2\times D_6$), $C_6\times A_4$ ($C_3\times D_4$)

Order 48: $C_2^3\times C_6$ ($C_6\times S_3$), $C_2^3\times C_6$ ($S_3^2$), $C_2^2\times D_6$ ($C_6\times S_3$), $C_2^2\times A_4$ ($C_6\times S_3$), $A_4:C_4$ ($C_6\times S_3$)

Order 36: $C_6^2$ ($C_2\times S_4$), $C_6^2$ ($C_6:D_4$), $C_6\times S_3$ ($C_2\times S_4$), $C_3\times A_4$ ($C_6\times D_4$), $C_3\times A_4$ ($C_2\times D_{12}$)

Order 24: $C_2^2\times C_6$ ($C_6\times D_6$), $C_2^2\times C_6$ ($S_3\times D_6$), $C_2^2\times C_6$ ($C_6\wr C_2$), $C_2^2\times C_6$ ($C_3:D_{12}$), $C_2\times D_6$ ($C_3\times S_4$), $C_2\times A_4$ ($C_6\times D_6$), $C_2\times A_4$ ($C_3\times D_{12}$)

Order 18: $C_3\times C_6$ ($C_2^2\times S_4$), $C_3\times C_6$ ($\GL(2,\mathbb{Z}/4)$)

Order 16: $C_2^4$ ($C_3\times S_3^2$)

Order 12: $C_2\times C_6$ ($C_6\times S_4$), $C_2\times C_6$ ($S_3\times S_4$), $C_2\times C_6$ ($C_6^2:C_2^2$), $C_2\times C_6$ ($C_6:D_{12}$), $D_6$ ($C_6\times S_4$), $A_4$ ($C_6\times D_{12}$)

Order 9: $C_3^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_6\times S_3^2$), $C_2^3$ ($C_3^2:D_{12}$)

Order 6: $C_6$ ($C_3\times \GL(2,\mathbb{Z}/4)$), $C_6$ ($D_6:S_4$), $C_6$ ($C_2\times C_6\times S_4$), $C_6$ ($D_6\times S_4$)

Order 4: $C_2^2$ ($C_6^2:D_6$), $C_2^2$ ($C_6^2.D_6$)

Order 3: $C_3$ ($C_6\times \GL(2,\mathbb{Z}/4)$), $C_3$ ($C_6:\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_6\times S_3\times S_4$), $C_2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($C_6\times D_6:S_4$)

Series

Derived series

$C_6\times D_6:S_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_6\times D_6:S_4$

$\rhd$

$C_6\times A_4\times D_6$

$\rhd$

$A_4\times C_6^2$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6\times D_6:S_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2\times C_6$

Character theory

Complex character table

See the $144 \times 144$ 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 $86 \times 86$ rational character table.