Abstract group 1119744.bp: $C_6^4.(C_6^2:C_4\times S_3)$

• Label: 1119744.bp
• Order: \( 2^{9} \cdot 3^{7} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^4.(C_6^2:C_4\times S_3)$
Order:	\(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	Group of order \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \)
Composition factors:	$C_2$ x 9, $C_3$ x 7
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	8	9	12	18	24
Elements	1	20799	6722	77760	239454	62208	12960	412128	69984	217728	1119744
Conjugacy classes	1	25	17	22	392	8	4	127	48	20	664
Divisions	1	25	17	18	388	4	4	117	36	10	620
Autjugacy classes	1	18	13	10	172	2	2	37	10	3	268

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid c^{6}=d^{6}=e^{6}=f^{6}=g^{6}=h^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,3,8,12,10,15)(2,6,13,17,14,18,4,9,16,5,11,7)(19,21,26,28)(20,23,22,25,29,24,27,30) \!\cdots\! \rangle$
Transitive group:	36T36525				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_6^5$ . $(C_6^2:C_4)$	$(C_6^4.C_6^2:C_4)$ . $S_3$	$C_6^4$ . $(C_6^2:C_4\times S_3)$	$C_3^4$ . $(A_4^2:D_6.D_4)$	all 58

Elements of the group are displayed as permutations of degree 30.

Homology

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

Subgroups

There are 95 normal subgroups (69 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^4.(A_4^2:C_4\times D_6)$
Commutator:	$G' \simeq$ $C_6^4.C_3^3.C_2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_3^3\times C_6$	$G/\Phi \simeq$ $A_4^2:C_4\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^5$	$G/\operatorname{Fit} \simeq$ $C_2\times C_3^2:C_4$
Radical:	$R \simeq$ $C_6^4.(C_6^2:C_4\times S_3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_3^4:(C_4\times D_6)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 18 (order at least 62208) are shown, as well as all normal subgroups of any index.

Order 1119744: $C_6^4.(C_6^2:C_4\times S_3)$

Order 559872: $C_3^4.(A_4^2:C_4\times D_6)$ x 2, $C_3^4.(D_6\times A_4^2):C_4$ x 2, $C_6^4.C_3^3.C_2^4$, $C_6^4.(C_3\times C_6^2):C_4$, $C_6^4.C_6^2:C_{12}$

Order 373248: $C_6^4.(C_2\times C_6^2:C_4)$

Order 279936: $C_6^4.C_3^3.C_2^3$ x 8, $C_3^4.(A_4^2:C_4\times S_3)$ x 4, $C_2\times C_3^5:(A_4^2:C_4)$ x 2, $C_2^5.\He_3^2:C_{12}$ x 2, $C_6^4.C_3^3.C_2^3$, $C_6^4.C_3^3.C_2^3$, $C_3^5.A_4^2.C_2^3$

Order 186624: $C_6^4.C_6^2:C_4$ x 2, $C_6^4.C_6^2:C_4$ x 2, $C_6^4.C_6^2:C_4$ x 2, $C_6^4.C_6^2.C_2^2$, $C_6^2.C_3^4.C_2^5.C_2$, $C_6^4.D_6^2$

Order 139968: $C_6^4.C_3^2.D_6$ x 10, $C_6^4.C_3^2.D_6$ x 4, $C_3^5.A_4^2.C_2^2$ x 4, $C_6^4.C_3^3.C_2^2$ x 3, $C_3^5:(A_4^2:C_4)$ x 2, $C_6^4.C_3^2:C_{12}$ x 2, $(C_2\times C_6^3).C_3^3.C_6.C_2$, $C_3^4.C_6^2.C_6.C_2^3$, $C_6^4.C_3^3.C_2^2$

Order 124416: $(C_3\times C_6^4).C_2^4.C_2$

Order 93312: $C_6^2.C_3^4.C_2^4.C_2$ x 24, $C_6^4.C_3^2.C_2^3$ x 9, $C_3^2.A_4^2.C_3.D_6.C_2$ x 4, $C_3^4:(C_2\times A_4^2:C_4)$ x 4, $C_2^5:(\He_3^2:C_4)$ x 4, $C_6^4.C_3.D_6.C_2$ x 3, $C_6^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2^5$, $C_6^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2^4.C_2$, $C_3^3.A_4^2.C_6.C_2^2$, $C_6^2.C_3^3.C_6.C_2^4$, $C_2^4.C_3^4.C_6^2.C_2$, $(C_3\times C_6^4).C_6.C_2^2$, $(C_2\times C_6^3).C_3^3.C_2^3$, $C_6^4.C_6^2.C_2$, $C_6^2.C_6^2.C_3^2.C_2^3$, $C_6^2.C_6^2.C_3.D_6.C_2$, $(C_6^2\times A_4).C_3^3.C_2^3$, $(C_3\times C_6^2).C_6^3.C_2^2$, $C_6^2.C_3^4.C_2^4.C_2$, $C_6^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2^4.C_2$, $C_6^4.C_6^2.C_2$, $C_3^5.A_4.C_2^5$

Order 69984: $C_3^4.C_6^2.C_6.C_2^2$ x 8, $C_3^4.C_6^2.D_6.C_2$ x 4, $C_3^5.A_4^2.C_2$ x 3, $C_6^4.C_3^3.C_2$ x 3, $C_6^4.C_3^3.C_2$ x 3, $C_6^4.C_3^3.C_2$ x 2, $C_6^2.C_3^4.C_6.C_2^2$, $C_3^4.C_6^2.C_6.C_2^2$, $C_3^4.C_6^2.D_6.C_2$, $\He_3^2:(D_6.D_4)$

Order 62208: $C_3^5.(C_2\times D_4).C_2^4$ x 4, $(C_3\times C_6^4).C_2^3.C_2$ x 4, $C_6^2.C_6^3.C_2^3$ x 3, $C_6^4.C_6.C_2^3$, $C_3^5.C_2^6.C_2^2$, $C_6^2.C_6^3.C_2^3$, $(C_2\times C_6^3).C_6^2.C_2^2$, $(C_3\times C_6^4).C_2^3.C_2$, $(C_2\times C_6^3).D_6^2$, $C_6^2.C_6^3.C_2^3$, $C_6^4.C_6.C_2^3$, $(C_2^3\times C_6).C_3^4.C_2^4$, $(C_3^4\times C_6).C_2^6.C_2$, $(C_3\times C_6^4).C_2^3.C_2$, $C_3^5.C_2^6.C_2^2$, $(C_2\times C_6^3).D_6^2$

Order 46656: $C_6^4.C_3.D_6$ x 2, $C_6^4.C_6^2$

Order 34992: $C_6^4.C_3^3$

Order 31104: $C_2^6\times C_3^2:(C_3^3:C_2)$

Order 23328: $C_6^4.C_3.S_3$ x 2, $C_6^4.C_3.C_6$

Order 15552: $C_2^5\times C_3^2:(C_3^3:C_2)$ x 2, $C_2\times C_6^5$

Order 11664: $C_6^4.C_3^2$

Order 7776: $C_2^4\times C_3^2:(C_3^3:C_2)$ x 2, $C_6^5$

Order 5184: $C_2^2\times C_6^4$

Order 3888: $C_3\times C_6^4$

Order 2592: $C_2\times C_6^4$

Order 2187: $C_3^4.C_3^3$

Order 1944: $C_3^5:C_2^3$

Order 1296: $C_6^4$

Order 972: $C_3^4:D_6$ x 2, $C_3^3\times C_6^2$

Order 576: $C_2^4\times C_6^2$

Order 512: $C_2^7:C_4$

Order 486: $C_3^4:S_3$ x 2, $C_3^4\times C_6$

Order 324: $C_3^2\times C_6^2$

Order 288: $C_2^3\times C_6^2$

Order 243: $C_3^5$

Order 192: $C_2^5\times C_6$ x 2

Order 162: $C_3^3\times C_6$

Order 144: $C_2^2\times C_6^2$

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 64: $C_2^6$

Order 48: $C_2^3\times C_6$ x 2

Order 36: $C_6^2$

Order 32: $C_2^5$

Order 18: $C_3\times C_6$

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 18 (order at least 62208) are shown, as well as all normal subgroups of any index.

Order 1119744: $C_6^4.(C_6^2:C_4\times S_3)$

Order 559872: $C_6^4.C_3^3.C_2^4$, $C_3^4.(A_4^2:C_4\times D_6)$, $C_3^4.(D_6\times A_4^2):C_4$, $C_6^4.(C_3\times C_6^2):C_4$, $C_6^4.C_6^2:C_{12}$

Order 373248: $C_6^4.(C_2\times C_6^2:C_4)$

Order 279936: $C_6^4.C_3^3.C_2^3$ x 5, $C_6^4.C_3^3.C_2^3$, $C_6^4.C_3^3.C_2^3$, $C_3^5.A_4^2.C_2^3$, $C_3^4.(A_4^2:C_4\times S_3)$, $C_2\times C_3^5:(A_4^2:C_4)$, $C_2^5.\He_3^2:C_{12}$

Order 186624: $C_6^4.C_6^2:C_4$ x 2, $C_6^4.C_6^2.C_2^2$, $C_6^2.C_3^4.C_2^5.C_2$, $C_6^4.D_6^2$, $C_6^4.C_6^2:C_4$, $C_6^4.C_6^2:C_4$

Order 139968: $C_6^4.C_3^2.D_6$ x 4, $C_6^4.C_3^2.D_6$ x 3, $C_3^5.A_4^2.C_2^2$ x 3, $C_6^4.C_3^3.C_2^2$ x 2, $(C_2\times C_6^3).C_3^3.C_6.C_2$, $C_3^4.C_6^2.C_6.C_2^3$, $C_6^4.C_3^3.C_2^2$, $C_3^5:(A_4^2:C_4)$, $C_6^4.C_3^2:C_{12}$

Order 124416: $(C_3\times C_6^4).C_2^4.C_2$

Order 93312: $C_6^2.C_3^4.C_2^4.C_2$ x 10, $C_6^4.C_3.D_6.C_2$ x 3, $C_6^4.C_3^2.C_2^3$ x 2, $C_2^5:(\He_3^2:C_4)$ x 2, $C_6^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2^5$, $C_6^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2^4.C_2$, $C_3^3.A_4^2.C_6.C_2^2$, $C_6^2.C_3^3.C_6.C_2^4$, $C_2^4.C_3^4.C_6^2.C_2$, $(C_3\times C_6^4).C_6.C_2^2$, $(C_2\times C_6^3).C_3^3.C_2^3$, $C_6^4.C_6^2.C_2$, $C_6^2.C_6^2.C_3^2.C_2^3$, $C_6^2.C_6^2.C_3.D_6.C_2$, $C_3^2.A_4^2.C_3.D_6.C_2$, $(C_6^2\times A_4).C_3^3.C_2^3$, $(C_3\times C_6^2).C_6^3.C_2^2$, $C_6^2.C_3^4.C_2^4.C_2$, $C_6^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2^4.C_2$, $C_6^4.C_6^2.C_2$, $C_3^5.A_4.C_2^5$, $C_3^4:(C_2\times A_4^2:C_4)$

Order 69984: $C_3^4.C_6^2.D_6.C_2$ x 4, $C_3^5.A_4^2.C_2$ x 2, $C_6^4.C_3^3.C_2$ x 2, $C_6^4.C_3^3.C_2$ x 2, $C_6^4.C_3^3.C_2$ x 2, $C_6^2.C_3^4.C_6.C_2^2$, $C_3^4.C_6^2.C_6.C_2^2$, $C_3^4.C_6^2.D_6.C_2$, $C_3^4.C_6^2.C_6.C_2^2$, $\He_3^2:(D_6.D_4)$

Order 62208: $C_3^5.(C_2\times D_4).C_2^4$ x 2, $(C_3\times C_6^4).C_2^3.C_2$ x 2, $C_6^4.C_6.C_2^3$, $C_3^5.C_2^6.C_2^2$, $C_6^2.C_6^3.C_2^3$, $(C_2\times C_6^3).C_6^2.C_2^2$, $(C_3\times C_6^4).C_2^3.C_2$, $(C_2\times C_6^3).D_6^2$, $C_6^2.C_6^3.C_2^3$, $C_6^4.C_6.C_2^3$, $(C_2^3\times C_6).C_3^4.C_2^4$, $(C_3^4\times C_6).C_2^6.C_2$, $C_6^2.C_6^3.C_2^3$, $(C_3\times C_6^4).C_2^3.C_2$, $C_3^5.C_2^6.C_2^2$, $(C_2\times C_6^3).D_6^2$

Order 46656: $C_6^4.C_3.D_6$ x 2, $C_6^4.C_6^2$

Order 34992: $C_6^4.C_3^3$

Order 31104: $C_2^6\times C_3^2:(C_3^3:C_2)$

Order 23328: $C_6^4.C_3.S_3$, $C_6^4.C_3.C_6$

Order 15552: $C_2^5\times C_3^2:(C_3^3:C_2)$ x 2, $C_2\times C_6^5$

Order 11664: $C_6^4.C_3^2$

Order 7776: $C_6^5$, $C_2^4\times C_3^2:(C_3^3:C_2)$

Order 5184: $C_2^2\times C_6^4$

Order 3888: $C_3\times C_6^4$

Order 2592: $C_2\times C_6^4$

Order 2187: $C_3^4.C_3^3$

Order 1944: $C_3^5:C_2^3$

Order 1296: $C_6^4$

Order 972: $C_3^4:D_6$ x 2, $C_3^3\times C_6^2$

Order 576: $C_2^4\times C_6^2$

Order 512: $C_2^7:C_4$

Order 486: $C_3^4\times C_6$, $C_3^4:S_3$

Order 324: $C_3^2\times C_6^2$

Order 288: $C_2^3\times C_6^2$

Order 243: $C_3^5$

Order 192: $C_2^5\times C_6$ x 2

Order 162: $C_3^3\times C_6$

Order 144: $C_2^2\times C_6^2$

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 64: $C_2^6$

Order 48: $C_2^3\times C_6$ x 2

Order 36: $C_6^2$

Order 32: $C_2^5$

Order 18: $C_3\times C_6$

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1119744: $C_6^4.(C_6^2:C_4\times S_3)$ ($C_1$)

Order 559872: $C_3^4.(A_4^2:C_4\times D_6)$ ($C_2$) x 2, $C_3^4.(D_6\times A_4^2):C_4$ ($C_2$) x 2, $C_6^4.C_3^3.C_2^4$ ($C_2$), $C_6^4.(C_3\times C_6^2):C_4$ ($C_2$), $C_6^4.C_6^2:C_{12}$ ($C_2$)

Order 279936: $C_6^4.C_3^3.C_2^3$ ($C_2^2$) x 2, $C_6^4.C_3^3.C_2^3$ ($C_4$) x 2, $C_2\times C_3^5:(A_4^2:C_4)$ ($C_2^2$) x 2, $C_2^5.\He_3^2:C_{12}$ ($C_2^2$) x 2, $C_6^4.C_3^3.C_2^3$ ($C_4$), $C_6^4.C_3^3.C_2^3$ ($C_2^2$), $C_3^5.A_4^2.C_2^3$ ($C_4$)

Order 186624: $C_6^4.C_6^2:C_4$ ($S_3$)

Order 139968: $C_6^4.C_3^2.D_6$ ($D_4$) x 4, $C_6^4.C_3^2.D_6$ ($C_2\times C_4$) x 2, $C_3^5.A_4^2.C_2^2$ ($C_2\times C_4$) x 2, $C_6^4.C_3^3.C_2^2$ ($C_2\times C_4$), $(C_2\times C_6^3).C_3^3.C_6.C_2$ ($C_2^3$), $C_6^4.C_3^3.C_2^2$ ($C_2\times C_4$)

Order 93312: $C_2^5:(\He_3^2:C_4)$ ($D_6$) x 2, $C_6^4.C_6^2.C_2$ ($D_6$)

Order 69984: $C_3^5.A_4^2.C_2$ ($C_2^2:C_4$) x 2, $C_6^4.C_3^3.C_2$ ($C_2\times D_4$) x 2, $C_6^4.C_3^3.C_2$ ($C_2^2:C_4$) x 2, $C_6^4.C_3^3.C_2$ ($C_2^2\times C_4$)

Order 46656: $C_6^4.C_3.D_6$ ($C_4\times S_3$), $C_6^4.C_3.D_6$ ($C_2\times D_6$), $C_6^4.C_6^2$ ($C_4\times S_3$)

Order 34992: $C_6^4.C_3^3$ ($C_2^3:C_4$)

Order 31104: $C_2^6\times C_3^2:(C_3^3:C_2)$ ($C_3^2:C_4$)

Order 23328: $C_6^4.C_3.S_3$ ($S_3\times D_4$) x 2, $C_6^4.C_3.C_6$ ($C_4\times D_6$)

Order 15552: $C_2^5\times C_3^2:(C_3^3:C_2)$ ($C_2\times C_3^2:C_4$) x 2, $C_2\times C_6^5$ ($C_2\times C_3^2:C_4$)

Order 11664: $C_6^4.C_3^2$ ($D_6.D_4$)

Order 7776: $C_2^4\times C_3^2:(C_3^3:C_2)$ ($C_6^2:C_4$) x 2, $C_6^5$ ($C_6^2:C_4$)

Order 5184: $C_2^2\times C_6^4$ ($C_3^2:C_4\times S_3$)

Order 3888: $C_3\times C_6^4$ ($C_2\times C_6^2:C_4$)

Order 2592: $C_2\times C_6^4$ ($C_3^2:C_4\times D_6$)

Order 1944: $C_3^5:C_2^3$ ($A_4^2:C_4$)

Order 1296: $C_6^4$ ($C_6^2:C_4\times S_3$)

Order 972: $C_3^4:D_6$ ($C_2\times A_4^2:C_4$) x 2, $C_3^3\times C_6^2$ ($C_2\times A_4^2:C_4$)

Order 576: $C_2^4\times C_6^2$ ($C_3^4:(C_4\times S_3)$)

Order 486: $C_3^4:S_3$ ($A_4^2.C_2^3.C_2$) x 2, $C_3^4\times C_6$ ($A_4^2.C_4.C_2^2$)

Order 324: $C_3^2\times C_6^2$ ($A_4^2:C_4\times S_3$)

Order 288: $C_2^3\times C_6^2$ ($C_3^4:(C_4\times D_6)$)

Order 243: $C_3^5$ ($C_2^3:(A_4^2:C_4)$)

Order 192: $C_2^5\times C_6$ ($C_3^4.C_3:S_3.C_2^2$), $C_2^5\times C_6$ ($C_3^4.C_3:S_3.C_2^2$)

Order 162: $C_3^3\times C_6$ ($A_4^2:C_4\times D_6$)

Order 144: $C_2^2\times C_6^2$ ($C_3^4:D_6.D_4$)

Order 96: $C_2^4\times C_6$ ($C_3^4.C_3:S_3.C_2^3$), $C_2^4\times C_6$ ($C_3^4.C_3:S_3.C_2^3$)

Order 81: $C_3^4$ ($A_4^2:D_6.D_4$)

Order 64: $C_2^6$ ($\He_3^2:(C_4\times S_3)$)

Order 48: $C_2^3\times C_6$ ($C_3^4.C_3^2.C_2^4.C_2$), $C_2^3\times C_6$ ($C_3^3.C_3^3.C_2^4.C_2$)

Order 36: $C_6^2$ ($(C_3^2\times A_4^2):(C_4\times S_3)$)

Order 32: $C_2^5$ ($\He_3^2:(C_4\times D_6)$)

Order 18: $C_3\times C_6$ ($(C_3^2\times A_4^2):(C_4\times D_6)$)

Order 16: $C_2^4$ ($\He_3^2:(D_6.D_4)$)

Order 12: $C_2\times C_6$ ($C_6^4.C_3:S_3.C_2^2$), $C_2\times C_6$ ($C_6^4.C_3:S_3.C_2^2$)

Order 9: $C_3^2$ ($(C_2^2\times C_6^3).C_6^2:C_4$)

Order 6: $C_6$ ($C_6^4.C_3:S_3.C_2^3$), $C_6$ ($C_6^4.C_3:S_3.C_2^3$)

Order 4: $C_2^2$ ($C_3^4.(A_4^2:C_4\times S_3)$)

Order 3: $C_3$ ($C_6^4.C_3^2.C_2^4.C_2$), $C_3$ ($C_6^4.C_3^2.C_2^4.C_2$)

Order 2: $C_2$ ($C_3^4.(A_4^2:C_4\times D_6)$)

Order 1: $C_1$ ($C_6^4.(C_6^2:C_4\times S_3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1119744: $C_6^4.(C_6^2:C_4\times S_3)$ ($C_1$)

Order 559872: $C_6^4.C_3^3.C_2^4$ ($C_2$), $C_3^4.(A_4^2:C_4\times D_6)$ ($C_2$), $C_3^4.(D_6\times A_4^2):C_4$ ($C_2$), $C_6^4.(C_3\times C_6^2):C_4$ ($C_2$), $C_6^4.C_6^2:C_{12}$ ($C_2$)

Order 279936: $C_6^4.C_3^3.C_2^3$ ($C_2^2$) x 2, $C_6^4.C_3^3.C_2^3$ ($C_4$) x 2, $C_6^4.C_3^3.C_2^3$ ($C_4$), $C_6^4.C_3^3.C_2^3$ ($C_2^2$), $C_3^5.A_4^2.C_2^3$ ($C_4$), $C_2\times C_3^5:(A_4^2:C_4)$ ($C_2^2$), $C_2^5.\He_3^2:C_{12}$ ($C_2^2$)

Order 186624: $C_6^4.C_6^2:C_4$ ($S_3$)

Order 139968: $C_6^4.C_3^2.D_6$ ($C_2\times C_4$) x 2, $C_3^5.A_4^2.C_2^2$ ($C_2\times C_4$) x 2, $C_6^4.C_3^3.C_2^2$ ($C_2\times C_4$), $(C_2\times C_6^3).C_3^3.C_6.C_2$ ($C_2^3$), $C_6^4.C_3^3.C_2^2$ ($C_2\times C_4$), $C_6^4.C_3^2.D_6$ ($D_4$)

Order 93312: $C_6^4.C_6^2.C_2$ ($D_6$), $C_2^5:(\He_3^2:C_4)$ ($D_6$)

Order 69984: $C_3^5.A_4^2.C_2$ ($C_2^2:C_4$), $C_6^4.C_3^3.C_2$ ($C_2\times D_4$), $C_6^4.C_3^3.C_2$ ($C_2^2:C_4$), $C_6^4.C_3^3.C_2$ ($C_2^2\times C_4$)

Order 46656: $C_6^4.C_3.D_6$ ($C_4\times S_3$), $C_6^4.C_3.D_6$ ($C_2\times D_6$), $C_6^4.C_6^2$ ($C_4\times S_3$)

Order 34992: $C_6^4.C_3^3$ ($C_2^3:C_4$)

Order 31104: $C_2^6\times C_3^2:(C_3^3:C_2)$ ($C_3^2:C_4$)

Order 23328: $C_6^4.C_3.S_3$ ($S_3\times D_4$), $C_6^4.C_3.C_6$ ($C_4\times D_6$)

Order 15552: $C_2^5\times C_3^2:(C_3^3:C_2)$ ($C_2\times C_3^2:C_4$) x 2, $C_2\times C_6^5$ ($C_2\times C_3^2:C_4$)

Order 11664: $C_6^4.C_3^2$ ($D_6.D_4$)

Order 7776: $C_6^5$ ($C_6^2:C_4$), $C_2^4\times C_3^2:(C_3^3:C_2)$ ($C_6^2:C_4$)

Order 5184: $C_2^2\times C_6^4$ ($C_3^2:C_4\times S_3$)

Order 3888: $C_3\times C_6^4$ ($C_2\times C_6^2:C_4$)

Order 2592: $C_2\times C_6^4$ ($C_3^2:C_4\times D_6$)

Order 1944: $C_3^5:C_2^3$ ($A_4^2:C_4$)

Order 1296: $C_6^4$ ($C_6^2:C_4\times S_3$)

Order 972: $C_3^4:D_6$ ($C_2\times A_4^2:C_4$) x 2, $C_3^3\times C_6^2$ ($C_2\times A_4^2:C_4$)

Order 576: $C_2^4\times C_6^2$ ($C_3^4:(C_4\times S_3)$)

Order 486: $C_3^4\times C_6$ ($A_4^2.C_4.C_2^2$), $C_3^4:S_3$ ($A_4^2.C_2^3.C_2$)

Order 324: $C_3^2\times C_6^2$ ($A_4^2:C_4\times S_3$)

Order 288: $C_2^3\times C_6^2$ ($C_3^4:(C_4\times D_6)$)

Order 243: $C_3^5$ ($C_2^3:(A_4^2:C_4)$)

Order 192: $C_2^5\times C_6$ ($C_3^4.C_3:S_3.C_2^2$), $C_2^5\times C_6$ ($C_3^4.C_3:S_3.C_2^2$)

Order 162: $C_3^3\times C_6$ ($A_4^2:C_4\times D_6$)

Order 144: $C_2^2\times C_6^2$ ($C_3^4:D_6.D_4$)

Order 96: $C_2^4\times C_6$ ($C_3^4.C_3:S_3.C_2^3$), $C_2^4\times C_6$ ($C_3^4.C_3:S_3.C_2^3$)

Order 81: $C_3^4$ ($A_4^2:D_6.D_4$)

Order 64: $C_2^6$ ($\He_3^2:(C_4\times S_3)$)

Order 48: $C_2^3\times C_6$ ($C_3^4.C_3^2.C_2^4.C_2$), $C_2^3\times C_6$ ($C_3^3.C_3^3.C_2^4.C_2$)

Order 36: $C_6^2$ ($(C_3^2\times A_4^2):(C_4\times S_3)$)

Order 32: $C_2^5$ ($\He_3^2:(C_4\times D_6)$)

Order 18: $C_3\times C_6$ ($(C_3^2\times A_4^2):(C_4\times D_6)$)

Order 16: $C_2^4$ ($\He_3^2:(D_6.D_4)$)

Order 12: $C_2\times C_6$ ($C_6^4.C_3:S_3.C_2^2$), $C_2\times C_6$ ($C_6^4.C_3:S_3.C_2^2$)

Order 9: $C_3^2$ ($(C_2^2\times C_6^3).C_6^2:C_4$)

Order 6: $C_6$ ($C_6^4.C_3:S_3.C_2^3$), $C_6$ ($C_6^4.C_3:S_3.C_2^3$)

Order 4: $C_2^2$ ($C_3^4.(A_4^2:C_4\times S_3)$)

Order 3: $C_3$ ($C_6^4.C_3^2.C_2^4.C_2$), $C_3$ ($C_6^4.C_3^2.C_2^4.C_2$)

Order 2: $C_2$ ($C_3^4.(A_4^2:C_4\times D_6)$)

Order 1: $C_1$ ($C_6^4.(C_6^2:C_4\times S_3)$)

Series

Derived series

$C_6^4.(C_6^2:C_4\times S_3)$

$\rhd$

$C_6^4.C_3^3.C_2$

$\rhd$

$C_6^4$

$\rhd$

$C_1$

Chief series

$C_6^4.(C_6^2:C_4\times S_3)$

$\rhd$

$C_6^4.C_3^3.C_2^4$

$\rhd$

$C_6^4.C_3^3.C_2^3$

$\rhd$

$(C_2\times C_6^3).C_3^3.C_6.C_2$

$\rhd$

$C_6^4.C_3^3.C_2$

$\rhd$

$C_6^4.C_3^3$

$\rhd$

$C_3\times C_6^4$

$\rhd$

$C_6^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^4.(C_6^2:C_4\times S_3)$

$\rhd$

$C_6^4.C_3^3.C_2$

$\rhd$

$C_6^4.C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

The $664 \times 664$ character table is not available for this group.

Rational character table

The $620 \times 620$ rational character table is not available for this group.