Abstract group 1728.46349: $C_2^3.S_3^3$

• Label: 1728.46349
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $18$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_2^3.S_3^3$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times S_4\times S_3\times C_2^2\times S_3$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
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	127	80	384	440	696	1728
Conjugacy classes	1	7	7	12	21	24	72
Divisions	1	7	7	10	18	14	57
Autjugacy classes	1	7	7	10	18	14	57

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	8	22	8	13	14	2	5	0	72
Irr. rational chars.	8	12	8	8	8	7	4	2	57

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$161280$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	7	9	11

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{4}=c^{6}=d^{6}=e^{6}=[a,d]=[b,c]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(6,7)(8,9,12,10)(11,13,15,14), (8,10,12,9)(11,13,15,14)(17,18), (2,4)(8,11) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.D_6)$ $\,\rtimes\,$ $S_4$	$A_4$ $\,\rtimes\,$ $(D_6.D_6)$	$(C_2.D_6^2)$ $\,\rtimes\,$ $S_3$	$(C_6^2:D_{12})$ $\,\rtimes\,$ $C_2$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $S_3^3$	$(C_6\times S_4)$ . $D_6$ (2)	$(C_6:S_4)$ . $D_6$	$(C_6.S_4)$ . $D_6$	all 26

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

Homology

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

Subgroups

There are 9500 subgroups in 814 conjugacy classes, 60 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_4\times S_3^2$
Commutator:	$G' \simeq$ $C_6^2:C_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_4\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2\times D_6$
Radical:	$R \simeq$ $C_2^3.S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1728: $C_2^3.S_3^3$

Order 864: $A_4\times C_6.D_6$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:(C_4\times S_4)$, $C_6.(S_3\times S_4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_3:C_{12}\times S_4$

Order 576: $(C_4\times S_3):S_4$, $(C_4\times S_3):S_4$, $C_2^2.D_6^2$

Order 432: $C_6^2:C_{12}$ x 2, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $(C_3^2\times A_4):C_4$, $C_3\times C_6.S_4$, $C_2.S_3^3$

Order 288: $D_6:S_4$ x 4, $C_2^3.S_3^2$ x 3, $C_4\times S_3\times A_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2.D_6^2$, $C_2.D_6^2$, $C_{12}:S_4$, $C_{12}.S_4$, $C_{12}:S_4$, $C_{12}\times S_4$, $D_6:S_4$, $C_2^3.S_3^2$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.C_2^3$, $C_6^2:D_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_6^2:Q_8$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 216: $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2\times S_4$, $C_3^2:D_{12}$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$

Order 192: $\GL(2,\mathbb{Z}/4):C_2$, $(C_4\times S_3):D_4$, $C_4^2:D_6$

Order 144: $C_6.S_4$ x 7, $C_6\times S_4$ x 5, $C_2^2.S_3^2$ x 4, $D_6.D_6$ x 4, $A_4\times C_3:C_4$ x 4, $C_6^2:C_4$ x 3, $C_6.D_{12}$ x 3, $A_4\times D_6$ x 3, $C_6^2:C_4$ x 3, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}\times A_4$ x 2, $C_6:D_{12}$ x 2, $A_4:C_{12}$ x 2, $D_6:C_{12}$, $C_{12}:C_{12}$, $C_6^2.C_2^2$, $C_{12}:C_{12}$, $C_2^2.S_3^2$, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_6^2:C_2^2$, $C_6:S_4$, $C_4:C_6^2$, $C_2^2.S_3^2$, $D_6.D_6$, $C_{12}.D_6$, $D_{12}:S_3$, $C_6^2:C_4$

Order 108: $C_3^2:D_6$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2\times D_6$, $C_3^2\times A_4$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 96: $D_6:D_4$ x 4, $D_6.D_4$ x 3, $C_4\times S_4$ x 3, $C_2^3.D_6$ x 2, $C_{12}:C_2^3$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $A_4:Q_8$ x 2, $C_2^3.D_6$ x 2, $C_{12}:D_4$ x 2, $(C_4\times S_3):C_4$, $D_6:D_4$, $C_2^2:D_{12}$, $C_2^3.D_6$, $C_4\times D_{12}$, $C_4^2:S_3$, $D_4:D_6$, $D_{12}:C_2^2$, $C_2^3:C_{12}$, $C_4:S_4$, $C_{12}:D_4$, $D_4\times C_{12}$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}.D_4$, $D_6:Q_8$, $D_6:Q_8$, $D_6.D_4$

Order 72: $C_6:C_{12}$ x 9, $C_6\times A_4$ x 6, $C_3\times S_4$ x 6, $C_3:D_{12}$ x 6, $C_6.D_6$ x 5, $C_6.D_6$ x 5, $C_6:D_6$ x 4, $C_6^2:C_2$ x 4, $C_6\wr C_2$ x 4, $C_3^2:Q_8$ x 4, $D_6:S_3$ x 4, $S_3\times A_4$ x 3, $S_3\times C_{12}$ x 3, $C_2\times C_6^2$ x 2, $D_4\times C_3^2$ x 2, $C_6.D_6$ x 2, $C_6\times D_6$, $C_3:S_4$, $C_6\times C_{12}$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3\times D_{12}$

Order 64: $C_4^2:C_2^2$

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

Order 48: $D_6:C_4$ x 12, $C_6:D_4$ x 10, $C_4\times D_6$ x 9, $C_6.D_4$ x 7, $A_4:C_4$ x 6, $C_6.D_4$ x 6, $C_6\times D_4$ x 5, $D_{12}:C_2$ x 5, $C_6.C_2^3$ x 4, $D_4:S_3$ x 4, $C_4\times A_4$ x 4, $C_2^2:C_{12}$ x 4, $C_{12}:C_4$ x 4, $C_2^2\times D_6$ x 3, $C_2\times S_4$ x 3, $C_2^2\times C_{12}$ x 3, $C_{12}:C_4$ x 3, $C_2\times D_{12}$ x 2, $C_6:Q_8$ x 2, $C_4:C_{12}$ x 2, $C_2^2\times A_4$, $C_4\times C_{12}$

Order 36: $C_3:C_{12}$ x 13, $C_6\times S_3$ x 11, $C_3^2:C_4$ x 9, $C_6:S_3$ x 8, $C_6^2$ x 6, $C_3\times A_4$ x 6, $C_3\times C_{12}$ x 3

Order 32: $C_4\times D_4$ x 4, $C_2^2.D_4$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $D_4:C_2^2$, $C_2^3\times C_4$, $C_4^2:C_2$

Order 27: $C_3^3$

Order 24: $C_3:D_4$ x 22, $C_6:C_4$ x 17, $C_4\times S_3$ x 15, $C_2\times C_{12}$ x 13, $C_2\times D_6$ x 13, $C_3\times D_4$ x 10, $C_2^2\times C_6$ x 7, $C_3:Q_8$ x 6, $D_{12}$ x 5, $C_2\times A_4$ x 5, $S_4$ x 4

Order 18: $C_3\times C_6$ x 10, $C_3\times S_3$ x 10, $C_3:S_3$ x 3

Order 16: $C_2^2:C_4$ x 10, $C_2^2\times C_4$ x 8, $C_2\times D_4$ x 7, $C_4:C_4$ x 6, $D_4:C_2$ x 4, $C_4^2$ x 2, $C_2^4$, $C_2\times Q_8$

Order 12: $D_6$ x 27, $C_2\times C_6$ x 22, $C_3:C_4$ x 19, $C_{12}$ x 14, $A_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 19, $D_4$ x 14, $C_2^3$ x 7, $Q_8$ x 2

Order 6: $C_6$ x 18, $S_3$ x 11

Order 4: $C_2^2$ x 16, $C_4$ x 10

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_2^3.S_3^3$

Order 864: $A_4\times C_6.D_6$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:(C_4\times S_4)$, $C_6.(S_3\times S_4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_3:C_{12}\times S_4$

Order 576: $(C_4\times S_3):S_4$, $(C_4\times S_3):S_4$, $C_2^2.D_6^2$

Order 432: $C_6^2:C_{12}$ x 2, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $(C_3^2\times A_4):C_4$, $C_3\times C_6.S_4$, $C_2.S_3^3$

Order 288: $D_6:S_4$ x 4, $C_2^3.S_3^2$ x 3, $C_4\times S_3\times A_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2.D_6^2$, $C_2.D_6^2$, $C_{12}:S_4$, $C_{12}.S_4$, $C_{12}:S_4$, $C_{12}\times S_4$, $D_6:S_4$, $C_2^3.S_3^2$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.C_2^3$, $C_6^2:D_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_6^2:Q_8$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 216: $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2\times S_4$, $C_3^2:D_{12}$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$

Order 192: $\GL(2,\mathbb{Z}/4):C_2$, $(C_4\times S_3):D_4$, $C_4^2:D_6$

Order 144: $C_6.S_4$ x 7, $C_6\times S_4$ x 5, $C_2^2.S_3^2$ x 4, $D_6.D_6$ x 4, $A_4\times C_3:C_4$ x 4, $C_6^2:C_4$ x 3, $C_6.D_{12}$ x 3, $A_4\times D_6$ x 3, $C_6^2:C_4$ x 3, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}\times A_4$ x 2, $C_6:D_{12}$ x 2, $A_4:C_{12}$ x 2, $D_6:C_{12}$, $C_{12}:C_{12}$, $C_6^2.C_2^2$, $C_{12}:C_{12}$, $C_2^2.S_3^2$, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_6^2:C_2^2$, $C_6:S_4$, $C_4:C_6^2$, $C_2^2.S_3^2$, $D_6.D_6$, $C_{12}.D_6$, $D_{12}:S_3$, $C_6^2:C_4$

Order 108: $C_3^2:D_6$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2\times D_6$, $C_3^2\times A_4$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 96: $D_6:D_4$ x 4, $D_6.D_4$ x 3, $C_4\times S_4$ x 3, $C_2^3.D_6$ x 2, $C_{12}:C_2^3$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $A_4:Q_8$ x 2, $C_2^3.D_6$ x 2, $C_{12}:D_4$ x 2, $(C_4\times S_3):C_4$, $D_6:D_4$, $C_2^2:D_{12}$, $C_2^3.D_6$, $C_4\times D_{12}$, $C_4^2:S_3$, $D_4:D_6$, $D_{12}:C_2^2$, $C_2^3:C_{12}$, $C_4:S_4$, $C_{12}:D_4$, $D_4\times C_{12}$, $C_2^3.D_6$, $C_{12}:D_4$, $C_{12}.D_4$, $D_6:Q_8$, $D_6:Q_8$, $D_6.D_4$

Order 72: $C_6:C_{12}$ x 9, $C_6\times A_4$ x 6, $C_3\times S_4$ x 5, $C_3:D_{12}$ x 5, $C_6.D_6$ x 5, $C_6.D_6$ x 5, $C_6:D_6$ x 4, $C_6\wr C_2$ x 4, $C_3^2:Q_8$ x 4, $D_6:S_3$ x 4, $S_3\times A_4$ x 3, $C_6^2:C_2$ x 3, $S_3\times C_{12}$ x 3, $C_2\times C_6^2$ x 2, $D_4\times C_3^2$ x 2, $C_6.D_6$ x 2, $C_6\times D_6$, $C_3:S_4$, $C_6\times C_{12}$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3\times D_{12}$

Order 64: $C_4^2:C_2^2$

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

Order 48: $D_6:C_4$ x 12, $C_6:D_4$ x 10, $C_4\times D_6$ x 9, $C_6.D_4$ x 7, $A_4:C_4$ x 6, $C_6.D_4$ x 6, $C_6\times D_4$ x 5, $D_{12}:C_2$ x 5, $C_6.C_2^3$ x 4, $D_4:S_3$ x 4, $C_4\times A_4$ x 4, $C_2^2:C_{12}$ x 4, $C_{12}:C_4$ x 4, $C_2^2\times D_6$ x 3, $C_2\times S_4$ x 3, $C_2^2\times C_{12}$ x 3, $C_{12}:C_4$ x 3, $C_2\times D_{12}$ x 2, $C_6:Q_8$ x 2, $C_4:C_{12}$ x 2, $C_2^2\times A_4$, $C_4\times C_{12}$

Order 36: $C_3:C_{12}$ x 13, $C_6\times S_3$ x 11, $C_3^2:C_4$ x 9, $C_6:S_3$ x 7, $C_6^2$ x 6, $C_3\times A_4$ x 6, $C_3\times C_{12}$ x 3

Order 32: $C_4\times D_4$ x 4, $C_2^2.D_4$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $D_4:C_2^2$, $C_2^3\times C_4$, $C_4^2:C_2$

Order 27: $C_3^3$

Order 24: $C_3:D_4$ x 18, $C_6:C_4$ x 17, $C_4\times S_3$ x 15, $C_2\times C_{12}$ x 13, $C_2\times D_6$ x 13, $C_3\times D_4$ x 10, $C_2^2\times C_6$ x 7, $C_3:Q_8$ x 6, $C_2\times A_4$ x 5, $D_{12}$ x 4, $S_4$ x 3

Order 18: $C_3\times C_6$ x 10, $C_3\times S_3$ x 9, $C_3:S_3$ x 3

Order 16: $C_2^2:C_4$ x 10, $C_2^2\times C_4$ x 8, $C_2\times D_4$ x 7, $C_4:C_4$ x 6, $D_4:C_2$ x 4, $C_4^2$ x 2, $C_2^4$, $C_2\times Q_8$

Order 12: $D_6$ x 24, $C_2\times C_6$ x 22, $C_3:C_4$ x 19, $C_{12}$ x 14, $A_4$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 19, $D_4$ x 12, $C_2^3$ x 7, $Q_8$ x 2

Order 6: $C_6$ x 18, $S_3$ x 10

Order 4: $C_2^2$ x 15, $C_4$ x 10

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_2^3.S_3^3$ ($C_1$)

Order 864: $A_4\times C_6.D_6$ ($C_2$), $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_3^2:(C_4\times S_4)$ ($C_2$), $C_6.(S_3\times S_4)$ ($C_2$), $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6^2:D_{12}$ ($C_2$), $C_3:C_{12}\times S_4$ ($C_2$)

Order 432: $C_6^2:C_{12}$ ($C_2^2$) x 2, $C_6:S_3\times A_4$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $A_4:C_6^2$ ($C_2^2$), $(C_3^2\times A_4):C_4$ ($C_2^2$), $C_3\times C_6.S_4$ ($C_2^2$)

Order 288: $C_2.D_6^2$ ($S_3$), $C_2^3.S_3^2$ ($S_3$), $C_2^3.S_3^2$ ($S_3$)

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

Order 144: $C_6\times S_4$ ($D_6$) x 2, $C_6^2:C_4$ ($D_6$) x 2, $A_4\times C_3:C_4$ ($D_6$) x 2, $C_6^2:C_2^2$ ($D_6$), $C_6:S_4$ ($D_6$), $C_6.S_4$ ($D_6$)

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

Order 72: $C_6\times A_4$ ($C_2\times D_6$) x 2, $C_2\times C_6^2$ ($C_2\times D_6$), $C_6.D_6$ ($S_4$)

Order 48: $C_6.C_2^3$ ($S_3^2$) x 2, $C_2\times S_4$ ($S_3^2$)

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

Order 24: $C_2^2\times C_6$ ($S_3\times D_6$) x 2, $C_2\times A_4$ ($S_3\times D_6$)

Order 18: $C_3\times C_6$ ($C_2^2\times S_4$)

Order 12: $C_3:C_4$ ($S_3\times S_4$) x 2, $C_2\times C_6$ ($C_{12}.D_6$), $C_2\times C_6$ ($D_{12}:S_3$), $A_4$ ($D_6.D_6$)

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

Order 8: $C_2^3$ ($S_3^3$)

Order 6: $C_6$ ($D_6\times S_4$) x 2

Order 4: $C_2^2$ ($C_2.S_3^3$)

Order 3: $C_3$ ($(C_4\times S_3):S_4$), $C_3$ ($(C_4\times S_3):S_4$)

Order 2: $C_2$ ($S_4\times S_3^2$)

Order 1: $C_1$ ($C_2^3.S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_2^3.S_3^3$ ($C_1$)

Order 864: $A_4\times C_6.D_6$ ($C_2$), $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_3^2:(C_4\times S_4)$ ($C_2$), $C_6.(S_3\times S_4)$ ($C_2$), $C_3^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6^2:D_{12}$ ($C_2$), $C_3:C_{12}\times S_4$ ($C_2$)

Order 432: $C_6^2:C_{12}$ ($C_2^2$) x 2, $C_6:S_3\times A_4$ ($C_2^2$), $C_6^2:D_6$ ($C_2^2$), $A_4:C_6^2$ ($C_2^2$), $(C_3^2\times A_4):C_4$ ($C_2^2$), $C_3\times C_6.S_4$ ($C_2^2$)

Order 288: $C_2.D_6^2$ ($S_3$), $C_2^3.S_3^2$ ($S_3$), $C_2^3.S_3^2$ ($S_3$)

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

Order 144: $C_6\times S_4$ ($D_6$) x 2, $C_6^2:C_4$ ($D_6$) x 2, $A_4\times C_3:C_4$ ($D_6$) x 2, $C_6^2:C_2^2$ ($D_6$), $C_6:S_4$ ($D_6$), $C_6.S_4$ ($D_6$)

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

Order 72: $C_6\times A_4$ ($C_2\times D_6$) x 2, $C_2\times C_6^2$ ($C_2\times D_6$), $C_6.D_6$ ($S_4$)

Order 48: $C_6.C_2^3$ ($S_3^2$) x 2, $C_2\times S_4$ ($S_3^2$)

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

Order 24: $C_2^2\times C_6$ ($S_3\times D_6$) x 2, $C_2\times A_4$ ($S_3\times D_6$)

Order 18: $C_3\times C_6$ ($C_2^2\times S_4$)

Order 12: $C_3:C_4$ ($S_3\times S_4$) x 2, $C_2\times C_6$ ($C_{12}.D_6$), $C_2\times C_6$ ($D_{12}:S_3$), $A_4$ ($D_6.D_6$)

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

Order 8: $C_2^3$ ($S_3^3$)

Order 6: $C_6$ ($D_6\times S_4$) x 2

Order 4: $C_2^2$ ($C_2.S_3^3$)

Order 3: $C_3$ ($(C_4\times S_3):S_4$), $C_3$ ($(C_4\times S_3):S_4$)

Order 2: $C_2$ ($S_4\times S_3^2$)

Order 1: $C_1$ ($C_2^3.S_3^3$)

Series

Derived series

$C_2^3.S_3^3$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^3.S_3^3$

$\rhd$

$C_6^2:D_{12}$

$\rhd$

$A_4:C_6^2$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^3.S_3^3$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_3^2\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $57 \times 57$ rational character table.