Abstract group 10368.hv: $C_6^3.(C_2\times S_4)$

• Label: 10368.hv
• Order: \( 2^{7} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^3.(C_2\times S_4)$
Order:	\(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_2^2\times C_6^2).C_3^3.C_2^5$, of order \(124416\)\(\medspace = 2^{9} \cdot 3^{5} \)
Composition factors:	$C_2$ x 7, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	247	26	1800	1238	864	3600	864	1728	10368
Conjugacy classes	1	9	5	14	79	6	18	6	6	144
Divisions	1	9	5	9	57	2	9	2	1	95
Autjugacy classes	1	8	5	7	54	2	7	2	1	87

Dimension	1	2	3	4	6	12	24
Irr. complex chars.	8	20	8	8	40	60	0	144
Irr. rational chars.	4	8	4	3	20	37	19	95

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$108$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{36}=c^{6}=d^{2}=e^{2}=f^{6}=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(2,3)(5,7,9,11)(6,8)(10,12)(14,15,17,20,23,22)(16,19)(18,21,24,27,26,29) \!\cdots\! \rangle$
Transitive group:	36T8415				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^3.S_4)$ $\,\rtimes\,$ $C_2$	$C_3^2$ $\,\rtimes\,$ $(C_2^5.D_{18})$	$(C_6^2.S_4)$ $\,\rtimes\,$ $(C_3:C_4)$	$(C_6^2.A_4)$ $\,\rtimes\,$ $(C_6:C_4)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^3.D_4)$ . $S_3$	$C_6^3$ . $(C_2\times S_4)$	$(C_6^2:C_{12})$ . $S_4$	$C_6^2$ . $(C_{12}.S_4)$	all 38

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

Homology

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

Subgroups

There are 62604 subgroups in 2890 conjugacy classes, 53 normal (49 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_2^2\times C_6^2):D_{18}$
Commutator:	$G' \simeq$ $(C_2^2\times C_6^2):C_9$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_6^2$	$G/\Phi \simeq$ $S_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6^3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6^3.(C_2\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $S_3\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2:C_9$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 10368: $C_6^3.(C_2\times S_4)$

Order 5184: $C_2\times C_2^4.C_3:D_9:C_3$, $C_6^3.S_4$, $(C_2^2\times C_6^2):C_{36}$

Order 3456: $C_6^3.C_4.C_2^2$, $C_6^2.(C_4\times S_4)$

Order 2592: $C_2^4.C_3:D_9:C_3$ x 2, $C_2\times C_2^4.C_3^2:C_9$, $C_6^3.D_6$

Order 1728: $C_3\times C_2^5.D_9$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_2\times C_6^2.S_4$, $C_6^3:D_4$, $C_6^3.C_2^3$, $(C_2^3\times C_6):C_{36}$, $C_6^3.D_4$

Order 1296: $C_6^3.S_3$ x 2, $C_6^2:D_{18}$ x 2, $C_6^2:C_{36}$, $(C_2^2\times C_6^2):C_9$

Order 1152: $C_6^2.(C_4\times D_4)$ x 2, $C_6^2.(C_4\times D_4)$, $C_2^5.D_{18}$

Order 864: $C_6^3:C_4$ x 8, $C_6^3.C_2^2$ x 6, $C_6^3.C_2^2$ x 4, $C_6^3:C_2^2$ x 2, $C_3\times C_2^4:C_{18}$ x 2, $C_6^2.S_4$ x 2, $C_6^2.D_{12}$ x 2, $C_2^2\times C_6^3$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6.(S_3\times S_4)$

Order 648: $C_6^2:D_9$ x 4, $C_6^2:C_{18}$ x 3, $C_3^2:(C_4\times D_9)$

Order 576: $(C_2^3\times C_6):C_{12}$ x 5, $C_6^2.C_4^2$ x 4, $C_6^2.C_2^4$ x 3, $C_2^2.D_6^2$ x 2, $C_2^5.D_9$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_2^4.S_3^2$ x 2, $C_6^2.C_4^2$ x 2, $C_2^4:D_{18}$, $C_6^2.C_2^4$, $C_2^2.D_6^2$, $C_2^4.S_3^2$, $C_2^4:C_{36}$

Order 432: $C_6^2:C_{12}$ x 19, $C_6^3:C_2$ x 16, $C_2\times C_6^3$ x 7, $C_6^2:C_{12}$ x 7, $C_3\times C_6.S_4$ x 5, $C_6^2.D_6$ x 2, $C_6^2.A_4$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^3:C_2$, $C_6^2.C_{12}$

Order 384: $C_2^5.D_6$ x 2, $C_2^5.D_6$

Order 324: $C_6^2:C_9$ x 3, $C_3^2:C_{36}$, $C_3^2:D_{18}$, $(C_3\times C_{18}).S_3$

Order 288: $C_6^2.D_4$ x 27, $C_6^2:D_4$ x 20, $C_6^2.D_4$ x 8, $C_6^2:D_4$ x 7, $C_6^2:D_4$ x 6, $C_2^3\times C_6^2$ x 5, $C_2^3.S_3^2$ x 4, $C_2^3.S_3^2$ x 4, $C_2^3.S_3^2$ x 4, $C_6^2.D_4$ x 4, $C_2^4:C_{18}$ x 2, $C_2^4:D_9$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_6^2.D_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_{12}.S_4$

Order 216: $C_6^3$ x 25, $C_6^2.C_6$ x 18, $C_6^2:C_6$ x 16, $C_6^2.C_6$ x 7, $C_6^2:C_6$ x 6, $C_3^2.S_4$ x 4, $C_6.S_3^2$ x 2, $C_6.D_{18}$

Order 192: $C_2^4.D_6$ x 5, $C_2^3.D_{12}$ x 4, $C_2^4:D_6$ x 3, $C_2^3.D_{12}$ x 2, $C_2^4:C_{12}$ x 2, $C_2^3.D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$, $C_2^5:C_6$, $C_2^4.D_6$

Order 162: $C_3^2:D_9$ x 2, $C_3^2:C_{18}$

Order 144: $C_6^2:C_4$ x 66, $C_6^2:C_2^2$ x 56, $C_2^2\times C_6^2$ x 55, $C_6^2:C_4$ x 22, $C_6^2:C_4$ x 19, $C_6^2:C_2^2$ x 16, $C_6^2:C_4$ x 7, $C_3^2:C_4^2$ x 6, $C_6.S_4$ x 5, $C_6.D_{12}$ x 4, $C_6.D_{12}$ x 4, $D_6.D_6$ x 4, $C_6^2:C_2^2$ x 3, $C_2^2.S_3^2$ x 2, $C_6.D_{12}$ x 2, $C_2^2.S_3^2$ x 2, $C_2^4:C_9$ x 2, $C_2^2:D_{18}$ x 2, $C_2^2:C_{36}$, $C_6^2:C_2^2$

Order 128: $C_2^4.D_4$

Order 108: $C_3\times C_6^2$ x 25, $C_3^2:C_{12}$ x 9, $C_3^2:D_6$ x 7, $C_3^2.A_4$ x 7, $C_9:C_{12}$ x 2, $C_3:C_{36}$, $C_3:D_{18}$

Order 96: $C_2^3.D_6$ x 28, $C_2^4:S_3$ x 20, $C_2^3:C_{12}$ x 8, $C_2^3:D_6$ x 7, $C_2^4:C_6$ x 6, $C_2^4\times C_6$ x 5, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 4, $C_{12}:D_4$ x 4, $C_2^2.D_{12}$ x 4, $D_6.D_4$ x 2, $C_2^3.D_6$ x 2, $C_2^3.D_6$ x 2, $C_{12}:C_2^3$ x 2, $C_2^3.D_6$

Order 81: $C_3^2:C_9$

Order 72: $C_2\times C_6^2$ x 227, $C_6:C_{12}$ x 64, $C_6\wr C_2$ x 56, $C_6\times D_6$ x 20, $C_6.D_6$ x 18, $C_6^2:C_2$ x 16, $C_2^2:C_{18}$ x 7, $C_6:D_6$ x 6, $C_6.D_6$ x 4, $C_2^2:D_9$ x 4, $C_6.D_6$ x 2, $C_4\times D_9$

Order 64: $C_2^4:C_4$ x 2, $C_2^2.C_4^2$ x 2, $C_2^4:C_4$, $C_2^3:D_4$, $C_4^2:C_2^2$

Order 54: $C_3^2\times C_6$ x 7, $C_3\times C_{18}$ x 2, $C_3:D_9$ x 2, $C_3^2:C_6$ x 2

Order 48: $C_6.D_4$ x 68, $C_2^3\times C_6$ x 56, $C_6:D_4$ x 56, $C_6.C_2^3$ x 24, $C_2^2:C_{12}$ x 19, $C_6\times D_4$ x 16, $C_2^2\times C_{12}$ x 7, $C_{12}:C_4$ x 6, $C_4\times D_6$ x 4, $D_6:C_4$ x 4, $C_6.D_4$ x 4, $C_2^2\times D_6$ x 3, $C_{12}:C_4$ x 2

Order 36: $C_6^2$ x 227, $C_3:C_{12}$ x 29, $C_6\times S_3$ x 24, $C_3^2:C_4$ x 9, $C_2^2:C_9$ x 7, $C_6:S_3$ x 7, $C_9:C_4$ x 2, $D_{18}$, $C_{36}$

Order 32: $C_2^3:C_4$ x 9, $C_2^2\wr C_2$ x 6, $C_4\times D_4$ x 4, $C_2^3:C_4$ x 2, $C_2^2\times D_4$ x 2, $C_2^5$, $C_4^2:C_2$

Order 27: $C_3\times C_9$ x 2, $C_3^3$

Order 24: $C_2^2\times C_6$ x 233, $C_6:C_4$ x 66, $C_3:D_4$ x 56, $C_2\times D_6$ x 20, $C_2\times C_{12}$ x 18, $C_3\times D_4$ x 16, $C_4\times S_3$ x 4

Order 18: $C_3\times C_6$ x 55, $C_3\times S_3$ x 6, $C_3:S_3$ x 2, $C_{18}$ x 2, $D_9$ x 2

Order 16: $C_2^2:C_4$ x 21, $C_2\times D_4$ x 16, $C_2^2\times C_4$ x 9, $C_2^4$ x 8, $C_4:C_4$ x 2, $C_4^2$ x 2

Order 12: $C_2\times C_6$ x 234, $C_3:C_4$ x 29, $D_6$ x 24, $C_{12}$ x 9

Order 9: $C_3^2$ x 5, $C_9$ x 2

Order 8: $C_2^3$ x 31, $C_2\times C_4$ x 20, $D_4$ x 16

Order 6: $C_6$ x 57, $S_3$ x 6

Order 4: $C_2^2$ x 32, $C_4$ x 9

Order 3: $C_3$ x 5

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 10368: $C_6^3.(C_2\times S_4)$

Order 5184: $C_2\times C_2^4.C_3:D_9:C_3$, $C_6^3.S_4$, $(C_2^2\times C_6^2):C_{36}$

Order 3456: $C_6^3.C_4.C_2^2$, $C_6^2.(C_4\times S_4)$

Order 2592: $C_2^4.C_3:D_9:C_3$, $C_2\times C_2^4.C_3^2:C_9$, $C_6^3.D_6$

Order 1728: $C_3\times C_2^5.D_9$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_2\times C_6^2.S_4$, $C_6^3:D_4$, $C_6^3.C_2^3$, $(C_2^3\times C_6):C_{36}$, $C_6^3.D_4$

Order 1296: $C_6^3.S_3$ x 2, $C_6^2:D_{18}$ x 2, $C_6^2:C_{36}$, $(C_2^2\times C_6^2):C_9$

Order 1152: $C_6^2.(C_4\times D_4)$ x 2, $C_6^2.(C_4\times D_4)$, $C_2^5.D_{18}$

Order 864: $C_6^3:C_4$ x 8, $C_6^3.C_2^2$ x 4, $C_6^3.C_2^2$ x 3, $C_6^3:C_2^2$ x 2, $C_3\times C_2^4:C_{18}$ x 2, $C_2^2\times C_6^3$, $C_6^2.S_4$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6.(S_3\times S_4)$, $C_6^2.D_{12}$

Order 648: $C_6^2:C_{18}$ x 3, $C_6^2:D_9$ x 2, $C_3^2:(C_4\times D_9)$

Order 576: $(C_2^3\times C_6):C_{12}$ x 5, $C_6^2.C_4^2$ x 4, $C_6^2.C_2^4$ x 3, $C_2^2.D_6^2$ x 2, $C_2^5.D_9$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_2^4.S_3^2$ x 2, $C_6^2.C_4^2$ x 2, $C_2^4:D_{18}$, $C_6^2.C_2^4$, $C_2^2.D_6^2$, $C_2^4.S_3^2$, $C_2^4:C_{36}$

Order 432: $C_6^2:C_{12}$ x 16, $C_6^3:C_2$ x 12, $C_2\times C_6^3$ x 7, $C_6^2:C_{12}$ x 7, $C_3\times C_6.S_4$ x 4, $C_6^2.D_6$ x 2, $C_6^2.A_4$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^3:C_2$, $C_6^2.C_{12}$

Order 384: $C_2^5.D_6$ x 2, $C_2^5.D_6$

Order 324: $C_6^2:C_9$ x 3, $C_3^2:C_{36}$, $C_3^2:D_{18}$, $(C_3\times C_{18}).S_3$

Order 288: $C_6^2.D_4$ x 24, $C_6^2:D_4$ x 9, $C_6^2.D_4$ x 8, $C_6^2:D_4$ x 6, $C_2^3\times C_6^2$ x 5, $C_2^3.S_3^2$ x 4, $C_2^3.S_3^2$ x 4, $C_2^3.S_3^2$ x 4, $C_6^2:D_4$ x 3, $C_2^4:C_{18}$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_6^2.D_4$ x 2, $C_6^2:D_4$ x 2, $C_2^4:D_9$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_{12}.S_4$, $C_6^2.D_4$

Order 216: $C_6^3$ x 25, $C_6^2.C_6$ x 16, $C_6^2:C_6$ x 8, $C_6^2.C_6$ x 6, $C_6^2:C_6$ x 5, $C_3^2.S_4$ x 2, $C_6.S_3^2$ x 2, $C_6.D_{18}$

Order 192: $C_2^4.D_6$ x 5, $C_2^3.D_{12}$ x 4, $C_2^4:D_6$ x 3, $C_2^3.D_{12}$ x 2, $C_2^4:C_{12}$ x 2, $C_2^3.D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$, $C_2^5:C_6$, $C_2^4.D_6$

Order 162: $C_3^2:D_9$, $C_3^2:C_{18}$

Order 144: $C_6^2:C_4$ x 48, $C_2^2\times C_6^2$ x 47, $C_6^2:C_2^2$ x 36, $C_6^2:C_4$ x 20, $C_6^2:C_4$ x 16, $C_6^2:C_2^2$ x 12, $C_6^2:C_4$ x 7, $C_3^2:C_4^2$ x 6, $C_6.D_{12}$ x 4, $C_6.D_{12}$ x 4, $C_6.S_4$ x 4, $D_6.D_6$ x 4, $C_6^2:C_2^2$ x 3, $C_2^2.S_3^2$ x 2, $C_6.D_{12}$ x 2, $C_2^2.S_3^2$ x 2, $C_2^4:C_9$ x 2, $C_2^2:D_{18}$ x 2, $C_2^2:C_{36}$, $C_6^2:C_2^2$

Order 128: $C_2^4.D_4$

Order 108: $C_3\times C_6^2$ x 25, $C_3^2:C_{12}$ x 7, $C_3^2.A_4$ x 6, $C_3^2:D_6$ x 5, $C_9:C_{12}$ x 2, $C_3:C_{36}$, $C_3:D_{18}$

Order 96: $C_2^3.D_6$ x 25, $C_2^4:S_3$ x 9, $C_2^3:C_{12}$ x 8, $C_2^3:D_6$ x 6, $C_2^4\times C_6$ x 5, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 4, $C_{12}:D_4$ x 4, $C_2^4:C_6$ x 3, $D_6.D_4$ x 2, $C_2^3.D_6$ x 2, $C_{12}:C_2^3$ x 2, $C_2^2.D_{12}$ x 2, $C_2^3.D_6$, $C_2^3.D_6$

Order 81: $C_3^2:C_9$

Order 72: $C_2\times C_6^2$ x 181, $C_6:C_{12}$ x 49, $C_6\wr C_2$ x 24, $C_6.D_6$ x 16, $C_6\times D_6$ x 15, $C_6^2:C_2$ x 8, $C_2^2:C_{18}$ x 6, $C_6:D_6$ x 5, $C_6.D_6$ x 4, $C_6.D_6$ x 2, $C_2^2:D_9$ x 2, $C_4\times D_9$

Order 64: $C_2^4:C_4$ x 2, $C_2^2.C_4^2$ x 2, $C_2^4:C_4$, $C_2^3:D_4$, $C_4^2:C_2^2$

Order 54: $C_3^2\times C_6$ x 7, $C_3\times C_{18}$ x 2, $C_3:D_9$, $C_3^2:C_6$

Order 48: $C_6.D_4$ x 50, $C_2^3\times C_6$ x 48, $C_6:D_4$ x 36, $C_6.C_2^3$ x 22, $C_2^2:C_{12}$ x 16, $C_6\times D_4$ x 12, $C_2^2\times C_{12}$ x 7, $C_{12}:C_4$ x 6, $C_4\times D_6$ x 4, $D_6:C_4$ x 4, $C_6.D_4$ x 4, $C_2^2\times D_6$ x 3, $C_{12}:C_4$ x 2

Order 36: $C_6^2$ x 181, $C_3:C_{12}$ x 20, $C_6\times S_3$ x 15, $C_3^2:C_4$ x 7, $C_2^2:C_9$ x 6, $C_6:S_3$ x 5, $C_9:C_4$ x 2, $D_{18}$, $C_{36}$

Order 32: $C_2^3:C_4$ x 9, $C_4\times D_4$ x 4, $C_2^2\wr C_2$ x 3, $C_2^2\times D_4$ x 2, $C_2^3:C_4$, $C_2^5$, $C_4^2:C_2$

Order 27: $C_3\times C_9$ x 2, $C_3^3$

Order 24: $C_2^2\times C_6$ x 186, $C_6:C_4$ x 51, $C_3:D_4$ x 24, $C_2\times C_{12}$ x 16, $C_2\times D_6$ x 15, $C_3\times D_4$ x 8, $C_4\times S_3$ x 4

Order 18: $C_3\times C_6$ x 47, $C_3\times S_3$ x 3, $C_{18}$ x 2, $C_3:S_3$, $D_9$

Order 16: $C_2^2:C_4$ x 18, $C_2\times D_4$ x 12, $C_2^2\times C_4$ x 9, $C_2^4$ x 8, $C_4:C_4$ x 2, $C_4^2$ x 2

Order 12: $C_2\times C_6$ x 186, $C_3:C_4$ x 20, $D_6$ x 15, $C_{12}$ x 7

Order 9: $C_3^2$ x 5, $C_9$ x 2

Order 8: $C_2^3$ x 30, $C_2\times C_4$ x 18, $D_4$ x 8

Order 6: $C_6$ x 48, $S_3$ x 3

Order 4: $C_2^2$ x 30, $C_4$ x 7

Order 3: $C_3$ x 5

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 10368: $C_6^3.(C_2\times S_4)$ ($C_1$)

Order 5184: $C_2\times C_2^4.C_3:D_9:C_3$ ($C_2$), $C_6^3.S_4$ ($C_2$), $(C_2^2\times C_6^2):C_{36}$ ($C_2$)

Order 2592: $C_2^4.C_3:D_9:C_3$ ($C_4$) x 2, $C_2\times C_2^4.C_3^2:C_9$ ($C_2^2$)

Order 1728: $C_2\times C_6^2.S_4$ ($S_3$), $C_6^3.D_4$ ($S_3$)

Order 1296: $(C_2^2\times C_6^2):C_9$ ($C_2\times C_4$)

Order 864: $C_6^2.S_4$ ($C_3:C_4$) x 2, $C_2^2\times C_6^3$ ($D_6$), $C_3\times C_2^4:C_{18}$ ($D_6$)

Order 576: $(C_2^2\times C_6^2):C_4$ ($D_9$)

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

Order 288: $C_2^3\times C_6^2$ ($D_{18}$), $C_2^3\times C_6^2$ ($S_3^2$)

Order 216: $C_6^3$ ($C_2\times S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_4\times D_9$), $C_2^2\times C_6^2$ ($C_6.D_6$), $C_6^2:C_4$ ($C_2^2:D_9$)

Order 108: $C_3\times C_6^2$ ($C_4\times S_4$)

Order 96: $C_2^4\times C_6$ ($C_3^2:D_6$), $C_2^4\times C_6$ ($S_3\times D_9$)

Order 72: $C_2\times C_6^2$ ($S_3\times S_4$), $C_2\times C_6^2$ ($C_2^2:D_{18}$)

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

Order 48: $C_2^3\times C_6$ ($C_6.S_3^2$), $C_2^3\times C_6$ ($C_6.D_{18}$)

Order 36: $C_6^2$ ($C_2^3.S_3^2$), $C_6^2$ ($C_{12}.S_4$)

Order 32: $C_2^5$ ($C_3^2:D_{18}$)

Order 27: $C_3^3$ ($C_2^5.D_6$)

Order 24: $C_2^2\times C_6$ ($C_6^2:D_6$), $C_2^2\times C_6$ ($C_6^2.D_6$)

Order 18: $C_3\times C_6$ ($C_2^4:S_3^2$), $C_3\times C_6$ ($C_2^4:D_{18}$)

Order 16: $C_2^4$ ($C_3^2:(C_4\times D_9)$)

Order 12: $C_2\times C_6$ ($C_3^2:(C_4\times S_4)$), $C_2\times C_6$ ($C_6.(S_3\times S_4)$)

Order 9: $C_3^2$ ($C_2^5.S_3^2$), $C_3^2$ ($C_2^5.D_{18}$)

Order 8: $C_2^3$ ($C_6^2:D_{18}$)

Order 6: $C_6$ ($(C_2^2\times C_6^2):D_6$), $C_6$ ($C_2^4:(S_3\times D_9)$)

Order 4: $C_2^2$ ($C_6^3.D_6$)

Order 3: $C_3$ ($C_2^4.(C_6\times D_9).C_2$), $C_3$ ($C_6^2.(C_4\times S_4)$)

Order 2: $C_2$ ($(C_2^2\times C_6^2):D_{18}$)

Order 1: $C_1$ ($C_6^3.(C_2\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 10368: $C_6^3.(C_2\times S_4)$ ($C_1$)

Order 5184: $C_2\times C_2^4.C_3:D_9:C_3$ ($C_2$), $C_6^3.S_4$ ($C_2$), $(C_2^2\times C_6^2):C_{36}$ ($C_2$)

Order 2592: $C_2^4.C_3:D_9:C_3$ ($C_4$), $C_2\times C_2^4.C_3^2:C_9$ ($C_2^2$)

Order 1728: $C_2\times C_6^2.S_4$ ($S_3$), $C_6^3.D_4$ ($S_3$)

Order 1296: $(C_2^2\times C_6^2):C_9$ ($C_2\times C_4$)

Order 864: $C_2^2\times C_6^3$ ($D_6$), $C_3\times C_2^4:C_{18}$ ($D_6$), $C_6^2.S_4$ ($C_3:C_4$)

Order 576: $(C_2^2\times C_6^2):C_4$ ($D_9$)

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

Order 288: $C_2^3\times C_6^2$ ($D_{18}$), $C_2^3\times C_6^2$ ($S_3^2$)

Order 216: $C_6^3$ ($C_2\times S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_4\times D_9$), $C_2^2\times C_6^2$ ($C_6.D_6$), $C_6^2:C_4$ ($C_2^2:D_9$)

Order 108: $C_3\times C_6^2$ ($C_4\times S_4$)

Order 96: $C_2^4\times C_6$ ($C_3^2:D_6$), $C_2^4\times C_6$ ($S_3\times D_9$)

Order 72: $C_2\times C_6^2$ ($S_3\times S_4$), $C_2\times C_6^2$ ($C_2^2:D_{18}$)

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

Order 48: $C_2^3\times C_6$ ($C_6.S_3^2$), $C_2^3\times C_6$ ($C_6.D_{18}$)

Order 36: $C_6^2$ ($C_2^3.S_3^2$), $C_6^2$ ($C_{12}.S_4$)

Order 32: $C_2^5$ ($C_3^2:D_{18}$)

Order 27: $C_3^3$ ($C_2^5.D_6$)

Order 24: $C_2^2\times C_6$ ($C_6^2:D_6$), $C_2^2\times C_6$ ($C_6^2.D_6$)

Order 18: $C_3\times C_6$ ($C_2^4:S_3^2$), $C_3\times C_6$ ($C_2^4:D_{18}$)

Order 16: $C_2^4$ ($C_3^2:(C_4\times D_9)$)

Order 12: $C_2\times C_6$ ($C_3^2:(C_4\times S_4)$), $C_2\times C_6$ ($C_6.(S_3\times S_4)$)

Order 9: $C_3^2$ ($C_2^5.S_3^2$), $C_3^2$ ($C_2^5.D_{18}$)

Order 8: $C_2^3$ ($C_6^2:D_{18}$)

Order 6: $C_6$ ($(C_2^2\times C_6^2):D_6$), $C_6$ ($C_2^4:(S_3\times D_9)$)

Order 4: $C_2^2$ ($C_6^3.D_6$)

Order 3: $C_3$ ($C_2^4.(C_6\times D_9).C_2$), $C_3$ ($C_6^2.(C_4\times S_4)$)

Order 2: $C_2$ ($(C_2^2\times C_6^2):D_{18}$)

Order 1: $C_1$ ($C_6^3.(C_2\times S_4)$)

Series

Derived series

$C_6^3.(C_2\times S_4)$

$\rhd$

$C_6^3.(C_2\times S_4)$

$\rhd$

$(C_2^2\times C_6^2):C_9$

$\rhd$

$(C_2^2\times C_6^2):C_9$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_6^3.(C_2\times S_4)$

$\rhd$

$C_6^3.(C_2\times S_4)$

$\rhd$

$C_2\times C_2^4.C_3:D_9:C_3$

$\rhd$

$C_2\times C_2^4.C_3:D_9:C_3$

$\rhd$

$C_2\times C_2^4.C_3^2:C_9$

$\rhd$

$C_2\times C_2^4.C_3^2:C_9$

$\rhd$

$(C_2^2\times C_6^2):C_9$

$\rhd$

$(C_2^2\times C_6^2):C_9$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_6^2.A_4$

$\rhd$

$C_6^2.A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_6^3.(C_2\times S_4)$

$\rhd$

$C_6^3.(C_2\times S_4)$

$\rhd$

$(C_2^2\times C_6^2):C_9$

$\rhd$

$(C_2^2\times C_6^2):C_9$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

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 $95 \times 95$ rational character table.