Abstract group 93312.cl: $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$

• Label: 93312.cl
• Order: \( 2^{7} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{5} \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$
Order:	\(93312\)\(\medspace = 2^{7} \cdot 3^{6} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_6^4.C_3^5.D_6.C_2^4$, of order \(60466176\)\(\medspace = 2^{10} \cdot 3^{10} \)
Composition factors:	$C_2$ x 7, $C_3$ x 6
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	9	12	18
Elements	1	711	2834	4536	28206	5184	36288	15552	93312
Conjugacy classes	1	16	50	7	2594	6	56	18	2748
Divisions	1	16	33	7	1344	6	28	12	1447
Autjugacy classes	1	9	9	3	80	1	6	2	111

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{6}=d^{6}=e^{6}=f^{6}=g^{6}=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,3,6)(2,4,8)(9,10,13)(12,17,15)(18,19,22,26,24,20,25,23,21)(27,29)(28,30) \!\cdots\! \rangle$
Transitive group:	36T19477				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:	$C_6^5$ . $D_6$ (4)	$(C_2\times C_6^5)$ . $S_3$ (4)	$(C_3\times C_6^4)$ . $S_4$	$C_6^4$ . $(C_3:S_4)$ (2)	all 81

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{3} \times C_{6}$
Commutator length:	not computed

Subgroups

There are 369 normal subgroups (57 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2\times C_2^4.C_3\wr C_3.S_3$
Commutator:	$G' \simeq$ $C_6^4.C_3.C_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3^2\times C_6$	$G/\Phi \simeq$ $C_2\times C_6^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^5$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_3^3:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$

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 32 (order at least 2916) are shown, as well as all normal subgroups of any index.

Order 93312: $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$

Order 46656: $C_2\times C_6^4.C_3.C_6$, $(C_3^2\times C_6^3).S_4$, $(C_3^2\times C_6^3).S_4$

Order 31104: $C_6^4.S_4$ x 9, $C_6^4.S_4$ x 2, $C_3\times C_6^3.C_6.C_2^3$, $C_6^4.S_4$

Order 23328: $C_3^5.\GL(2,\mathbb{Z}/4)$ x 3, $C_6^4.C_3.C_6$, $C_2\times C_6^4.C_3^2$, $(C_3^3\times C_6^2).S_4$

Order 15552: $C_6\times (C_2\times C_6^3).C_6$ x 9, $C_6^4:D_6$ x 9, $C_3\times C_6^3.S_4$ x 9, $C_3\times C_6^3.C_6.C_2^2$ x 7, $C_6\times (C_3^2\times C_6^2).C_2^3$ x 7, $C_2\times C_2^4.C_3^2:C_9.C_6$ x 2, $(C_3\times C_6^3).S_4$ x 2, $C_6^4.D_6$ x 2, $C_2\times C_2^4.C_3^4:C_3.C_2$, $C_2\times C_6^5$, $(C_3\times C_6^3).S_4$, $C_6^4:D_6$

Order 11664: $C_2^2\times C_6^2.C_3^3.C_3$ x 4, $C_2\times C_3^5:S_4$ x 3, $(C_3^4\times C_6).S_4$ x 3, $C_6^4.C_3^2$

Order 10368: $C_3^2\times (C_2\times C_6^2).C_2^4$ x 9, $(C_2\times C_6^3):S_4$ x 9, $C_3\times C_6^2.\GL(2,\mathbb{Z}/4)$ x 6, $C_3\times C_6^3.C_2^4$ x 4, $C_6^3.C_6.C_2^3$ x 3, $C_3\times C_6^2:\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^4\times C_6^2):D_9$ x 2, $C_3\times C_6^3.C_2^4$, $(C_2\times C_6^3):S_4$

Order 7776: $C_6^4:S_3$ x 27, $C_6\times (C_3\times C_6^2).C_6.C_2$ x 18, $C_6^5$ x 15, $C_3\times C_3^3.(C_6\times D_4).C_2$ x 15, $C_3\times (C_3^2\times C_6^2).C_2^3$ x 13, $C_6\times C_2^4.C_3\wr C_3$ x 9, $C_3\times (C_2\times C_6^3).C_6$ x 9, $C_6^4:S_3$ x 9, $C_2\times C_6\times C_3^4:D_4$ x 7, $C_6^4.S_3$ x 6, $C_3\times C_3^2.(C_6\times C_6.D_4)$ x 3, $C_6^4:S_3$ x 3, $C_2^4.C_3^2:C_9.C_6$ x 2, $C_2\times C_2^4.(C_3^4.C_3)$ x 2, $C_6^4.S_3$ x 2, $C_2^4.C_3^4:C_3.C_2$, $C_2\times C_2^4.C_3^4:C_3$, $C_6^4:S_3$

Order 5832: $C_2\times C_6^2.C_3^3.C_3$ x 8, $C_3^5:S_4$ x 3, $C_6^2.C_3^3.C_6$, $(C_3^3\times C_6^2):S_3$

Order 5184: $C_3^2\times C_6^2.(C_2\times D_4)$ x 63, $C_3\times C_6\times (C_2^3\times C_6):S_3$ x 63, $C_2^2\times C_6^4$ x 30, $C_3\times C_6^3.C_2^3$ x 28, $C_6\times (C_3\times C_6^2).C_2^3$ x 28, $C_6^3.C_6.C_2^2$ x 21, $C_2\times C_3^3.(C_6\times D_4).C_2$ x 18, $(C_2\times C_6)\wr C_3$ x 18, $C_6^3:S_4$ x 9, $C_6^3.S_4$ x 9, $C_6\times C_3^2.C_2^4:S_3$ x 7, $C_3\times C_6^3.C_2^3$ x 7, $C_6\times C_2^4:C_9.C_6$ x 6, $C_6^3.S_4$ x 6, $C_6^3.S_4$ x 6, $C_6\times C_2^4:\He_3.C_2$ x 5, $C_2\times C_2^4.C_3^2:C_9.C_2$ x 4, $C_2\times (C_3^2\times C_6^2).C_2^3$ x 3, $C_6^3.S_4$ x 3, $C_6^3:S_4$ x 3, $C_6^3.S_4$ x 2, $C_6^3.S_4$ x 2, $C_6^3:S_4$, $C_6^3.S_4$

Order 3888: $C_3\times C_6^4$ x 128, $C_3\times (C_3\times C_6^2).C_6.C_2$ x 42, $C_6\times C_3^4:D_4$ x 42, $C_2\times C_6\times (C_3\times C_6^2):C_3$ x 36, $C_3\times C_6^3.S_3$ x 27, $C_3\times C_6^3:S_3$ x 27, $C_3\times C_2^4.C_3\wr C_3$ x 9, $C_2^2\times C_2^2:(C_3^4.C_3)$ x 8, $C_2\times C_6\times C_3\times C_3^3:C_4$ x 7, $C_6^2.(C_3\times C_6).C_6$ x 6, $C_2\times C_3^4.S_4$ x 6, $C_2^2\times C_2^2:(C_3^4:C_3)$ x 4, $C_2\times C_3^4:S_4$ x 3, $(C_3\times C_6^3).S_3$ x 3, $C_2^4.(C_3^4.C_3)$ x 2, $C_2^2\times C_6\times C_3\times C_3^3:C_2$, $C_2^4.C_3^4:C_3$

Order 3456: $C_3\times (C_2\times C_6^2).C_2^4$ x 40, $C_6^2.\GL(2,\mathbb{Z}/4)$ x 18, $C_6^3.C_2^4$ x 12, $C_3^2\times C_3:(C_2^6.C_2)$ x 12, $C_6^2:\GL(2,\mathbb{Z}/4)$ x 9, $C_6^2.\GL(2,\mathbb{Z}/4)$ x 6, $C_6^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_6^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_3\times (C_2\times C_6^2).C_2^4$, $C_6^3.C_2^4$

Order 2916: $C_6^2.C_3^3.C_3$ x 4, $C_2^2\times C_3^4.C_3^2$, $C_3^5:D_6$, $(C_3^4\times C_6).S_3$

Order 2592: $C_3\times C_2^4:C_9.C_6$ x 6, $C_2\times C_6^4$ x 4, $C_3\times C_2^4:\He_3.C_2$ x 3

Order 1944: $C_3^2\times C_6^3$ x 3

Order 1728: $C_2^3\times C_6^3$ x 5

Order 1296: $C_6^4$ x 16, $C_3\times C_6^2.A_4$ x 6, $C_3\times C_6^2:A_4$ x 3

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

Order 864: $C_2^2\times C_6^3$ x 5

Order 729: $C_3^5:C_3$

Order 648: $C_3\times C_6^3$ x 12

Order 576: $C_2^4\times C_6^2$ x 7

Order 486: $C_3^4\times C_6$

Order 432: $C_2\times C_6^3$ x 20

Order 324: $C_3^2\times C_6^2$ x 16

Order 288: $C_2^3\times C_6^2$ x 7

Order 243: $C_3^5$

Order 216: $C_6^3$ x 15

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

Order 162: $C_3^3\times C_6$ x 4

Order 144: $C_2^2\times C_6^2$ x 28

Order 128: $C_2^3\wr C_2$

Order 108: $C_3\times C_6^2$ x 20

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

Order 81: $C_3^4$ x 4

Order 72: $C_2\times C_6^2$ x 21

Order 64: $C_2^6$

Order 54: $C_3^2\times C_6$ x 5

Order 48: $C_2^3\times C_6$ x 8

Order 36: $C_6^2$ x 28

Order 32: $C_2^5$

Order 27: $C_3^3$ x 5

Order 24: $C_2^2\times C_6$ x 6

Order 18: $C_3\times C_6$ x 7

Order 16: $C_2^4$ x 4

Order 12: $C_2\times C_6$ x 8

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 3

Order 6: $C_6$ x 2

Order 4: $C_2^2$ x 4

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 32 (order at least 2916) are shown, as well as all normal subgroups of any index.

Order 93312: $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$

Order 46656: $C_2\times C_6^4.C_3.C_6$, $(C_3^2\times C_6^3).S_4$, $(C_3^2\times C_6^3).S_4$

Order 31104: $C_3\times C_6^3.C_6.C_2^3$, $C_6^4.S_4$, $C_6^4.S_4$, $C_6^4.S_4$

Order 23328: $C_6^4.C_3.C_6$, $C_2\times C_6^4.C_3^2$, $(C_3^3\times C_6^2).S_4$, $C_3^5.\GL(2,\mathbb{Z}/4)$

Order 15552: $C_3\times C_6^3.C_6.C_2^2$ x 3, $C_6\times (C_3^2\times C_6^2).C_2^3$ x 3, $C_6\times (C_2\times C_6^3).C_6$, $C_2\times C_2^4.C_3^4:C_3.C_2$, $C_2\times C_6^5$, $C_2\times C_2^4.C_3^2:C_9.C_6$, $(C_3\times C_6^3).S_4$, $C_6^4.D_6$, $C_6^4:D_6$, $C_3\times C_6^3.S_4$, $(C_3\times C_6^3).S_4$, $C_6^4:D_6$

Order 11664: $C_2^2\times C_6^2.C_3^3.C_3$ x 2, $C_6^4.C_3^2$, $C_2\times C_3^5:S_4$, $(C_3^4\times C_6).S_4$

Order 10368: $C_3\times C_6^3.C_2^4$ x 2, $C_6^3.C_6.C_2^3$, $C_3\times C_6^3.C_2^4$, $C_3^2\times (C_2\times C_6^2).C_2^4$, $(C_2\times C_6^3):S_4$, $(C_2\times C_6^3):S_4$, $(C_2^4\times C_6^2):D_9$, $C_3\times C_6^2.\GL(2,\mathbb{Z}/4)$, $C_3\times C_6^2:\GL(2,\mathbb{Z}/4)$

Order 7776: $C_6^5$ x 8, $C_6\times (C_3\times C_6^2).C_6.C_2$ x 5, $C_3\times C_3^3.(C_6\times D_4).C_2$ x 4, $C_3\times (C_3^2\times C_6^2).C_2^3$ x 4, $C_2\times C_6\times C_3^4:D_4$ x 3, $C_2^4.C_3^2:C_9.C_6$, $C_2\times C_2^4.(C_3^4.C_3)$, $C_3\times C_3^2.(C_6\times C_6.D_4)$, $C_6\times C_2^4.C_3\wr C_3$, $C_3\times (C_2\times C_6^3).C_6$, $C_2^4.C_3^4:C_3.C_2$, $C_2\times C_2^4.C_3^4:C_3$, $C_6^4.S_3$, $C_6^4.S_3$, $C_6^4:S_3$, $C_6^4:S_3$, $C_6^4:S_3$, $C_6^4:S_3$

Order 5832: $C_2\times C_6^2.C_3^3.C_3$ x 3, $C_6^2.C_3^3.C_6$, $(C_3^3\times C_6^2):S_3$, $C_3^5:S_4$

Order 5184: $C_2^2\times C_6^4$ x 8, $C_3\times C_6^3.C_2^3$ x 6, $C_6\times (C_3\times C_6^2).C_2^3$ x 6, $C_6\times C_2^4:\He_3.C_2$ x 3, $C_3^2\times C_6^2.(C_2\times D_4)$ x 3, $C_3\times C_6\times (C_2^3\times C_6):S_3$ x 3, $C_6\times C_3^2.C_2^4:S_3$ x 3, $C_6^3.C_6.C_2^2$ x 3, $C_3\times C_6^3.C_2^3$ x 3, $C_2\times C_3^3.(C_6\times D_4).C_2$ x 2, $C_2\times C_2^4.C_3^2:C_9.C_2$ x 2, $(C_2\times C_6)\wr C_3$ x 2, $C_2\times (C_3^2\times C_6^2).C_2^3$, $C_6\times C_2^4:C_9.C_6$, $C_6^3:S_4$, $C_6^3.S_4$, $C_6^3:S_4$, $C_6^3.S_4$, $C_6^3.S_4$, $C_6^3.S_4$, $C_6^3.S_4$, $C_6^3.S_4$, $C_6^3.S_4$, $C_6^3:S_4$

Order 3888: $C_3\times C_6^4$ x 35, $C_3\times (C_3\times C_6^2).C_6.C_2$ x 11, $C_6\times C_3^4:D_4$ x 11, $C_2\times C_6\times C_3\times C_3^3:C_4$ x 3, $C_2\times C_6\times (C_3\times C_6^2):C_3$ x 2, $C_2^2\times C_2^2:(C_3^4:C_3)$ x 2, $C_2^2\times C_2^2:(C_3^4.C_3)$ x 2, $C_6^2.(C_3\times C_6).C_6$, $C_2^2\times C_6\times C_3\times C_3^3:C_2$, $C_2^4.C_3^4:C_3$, $C_3\times C_2^4.C_3\wr C_3$, $C_2^4.(C_3^4.C_3)$, $C_2\times C_3^4.S_4$, $C_3\times C_6^3.S_3$, $C_3\times C_6^3:S_3$, $C_2\times C_3^4:S_4$, $(C_3\times C_6^3).S_3$

Order 3456: $C_3\times (C_2\times C_6^2).C_2^4$ x 4, $C_6^3.C_2^4$ x 3, $C_3^2\times C_3:(C_2^6.C_2)$ x 2, $C_3\times (C_2\times C_6^2).C_2^4$, $C_6^3.C_2^4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $C_6^2.\GL(2,\mathbb{Z}/4)$

Order 2916: $C_6^2.C_3^3.C_3$ x 2, $C_2^2\times C_3^4.C_3^2$, $C_3^5:D_6$, $(C_3^4\times C_6).S_3$

Order 2592: $C_2\times C_6^4$ x 2, $C_3\times C_2^4:\He_3.C_2$, $C_3\times C_2^4:C_9.C_6$

Order 1944: $C_3^2\times C_6^3$

Order 1728: $C_2^3\times C_6^3$ x 3

Order 1296: $C_6^4$ x 4, $C_3\times C_6^2:A_4$, $C_3\times C_6^2.A_4$

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

Order 864: $C_2^2\times C_6^3$ x 3

Order 729: $C_3^5:C_3$

Order 648: $C_3\times C_6^3$ x 2

Order 576: $C_2^4\times C_6^2$ x 3

Order 486: $C_3^4\times C_6$

Order 432: $C_2\times C_6^3$ x 6

Order 324: $C_3^2\times C_6^2$ x 4

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

Order 243: $C_3^5$

Order 216: $C_6^3$ x 3

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

Order 162: $C_3^3\times C_6$ x 2

Order 144: $C_2^2\times C_6^2$ x 6

Order 128: $C_2^3\wr C_2$

Order 108: $C_3\times C_6^2$ x 6

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

Order 81: $C_3^4$ x 2

Order 72: $C_2\times C_6^2$ x 3

Order 64: $C_2^6$

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

Order 48: $C_2^3\times C_6$ x 4

Order 36: $C_6^2$ x 6

Order 32: $C_2^5$

Order 27: $C_3^3$ x 3

Order 24: $C_2^2\times C_6$ x 2

Order 18: $C_3\times C_6$ x 3

Order 16: $C_2^4$ x 2

Order 12: $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 3

Order 8: $C_2^3$

Order 6: $C_6$ x 2

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 93312: $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 46656: $C_2\times C_6^4.C_3.C_6$ ($C_2$), $(C_3^2\times C_6^3).S_4$ ($C_2$), $(C_3^2\times C_6^3).S_4$ ($C_2$)

Order 23328: $C_6^4.C_3.C_6$ ($C_2^2$)

Order 15552: $C_6\times (C_2\times C_6^3).C_6$ ($S_3$) x 9, $C_2\times C_2^4.C_3^2:C_9.C_6$ ($S_3$) x 2, $C_2\times C_2^4.C_3^4:C_3.C_2$ ($S_3$), $C_2\times C_6^5$ ($S_3$)

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

Order 7776: $C_3\times (C_2\times C_6^3).C_6$ ($D_6$) x 9, $C_2^4.C_3^2:C_9.C_6$ ($D_6$) x 2, $C_6^5$ ($D_6$), $C_2^4.C_3^4:C_3.C_2$ ($D_6$)

Order 5184: $C_6\times C_2^4:C_9.C_6$ ($C_3:S_3$) x 6, $C_2^2\times C_6^4$ ($C_3:S_3$) x 4, $C_6\times C_2^4:\He_3.C_2$ ($C_3:S_3$) x 3

Order 3888: $C_3\times C_2^4.C_3\wr C_3$ ($C_3:D_4$) x 9, $C_3\times C_6^4$ ($S_4$) x 3, $C_2^4.(C_3^4.C_3)$ ($C_3:D_4$) x 2, $C_3\times C_6^4$ ($C_3:D_4$), $C_2^4.C_3^4:C_3$ ($C_3:D_4$)

Order 2592: $C_3\times C_2^4:C_9.C_6$ ($C_6:S_3$) x 6, $C_2\times C_6^4$ ($C_6:S_3$) x 4, $C_3\times C_2^4:\He_3.C_2$ ($C_6:S_3$) x 3

Order 1944: $C_3^2\times C_6^3$ ($C_2\times S_4$) x 3

Order 1728: $C_2^3\times C_6^3$ ($C_3^2:S_3$) x 4, $C_2^3\times C_6^3$ ($C_3^2:S_3$)

Order 1296: $C_6^4$ ($C_3:S_4$) x 12, $C_3\times C_6^2.A_4$ ($C_6^2:C_2$) x 6, $C_6^4$ ($C_6^2:C_2$) x 4, $C_3\times C_6^2:A_4$ ($C_6^2:C_2$) x 3

Order 972: $C_3^3\times C_6^2$ ($\GL(2,\mathbb{Z}/4)$) x 3, $C_3^3\times C_6^2$ ($C_2^2:S_4$)

Order 864: $C_2^2\times C_6^3$ ($C_3^2:D_6$) x 4, $C_2^2\times C_6^3$ ($C_3^2:D_6$)

Order 648: $C_3\times C_6^3$ ($C_6:S_4$) x 12

Order 576: $C_2^4\times C_6^2$ ($C_3^3:S_3$) x 4, $C_2^4\times C_6^2$ ($C_3^3:S_3$) x 3

Order 486: $C_3^4\times C_6$ ($C_2^3:S_4$)

Order 432: $C_2\times C_6^3$ ($C_3^2:S_4$) x 12, $C_2\times C_6^3$ ($C_6^2:S_3$) x 4, $C_2\times C_6^3$ ($C_3^2:S_4$) x 3, $C_2\times C_6^3$ ($C_6^2:S_3$)

Order 324: $C_3^2\times C_6^2$ ($C_3:\GL(2,\mathbb{Z}/4)$) x 12, $C_3^2\times C_6^2$ ($(C_2\times C_6):S_4$) x 4

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

Order 243: $C_3^5$ ($C_2\wr S_3$)

Order 216: $C_6^3$ ($C_6^2:D_6$) x 12, $C_6^3$ ($C_6^2:D_6$) x 3

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

Order 162: $C_3^3\times C_6$ ($(C_2^2\times C_6):S_4$) x 4

Order 144: $C_2^2\times C_6^2$ ($C_3^3:S_4$) x 12, $C_2^2\times C_6^2$ ($C_3^3:S_4$) x 9, $C_2^2\times C_6^2$ ($(C_3\times C_6^2):S_3$) x 4, $C_2^2\times C_6^2$ ($(C_3\times C_6^2):S_3$) x 3

Order 108: $C_3\times C_6^2$ ($C_6^2:S_4$) x 12, $C_3\times C_6^2$ ($C_6^2:S_4$) x 4, $C_3\times C_6^2$ ($C_6^2:S_4$) x 3, $C_3\times C_6^2$ ($C_6^2:S_4$)

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

Order 81: $C_3^4$ ($(C_2\times C_6):\GL(2,\mathbb{Z}/4)$) x 4

Order 72: $C_2\times C_6^2$ ($C_6^3:S_3$) x 12, $C_2\times C_6^2$ ($C_6^3:S_3$) x 9

Order 64: $C_2^6$ ($C_3^5:S_3$)

Order 54: $C_3^2\times C_6$ ($C_2\times C_6^2:S_4$) x 4, $C_3^2\times C_6$ ($C_2\times C_6^2:S_4$)

Order 48: $C_2^3\times C_6$ ($C_3^4:S_4$) x 3, $C_2^3\times C_6$ ($C_3^4:S_4$) x 3, $C_2^3\times C_6$ ($(C_3^2\times C_6^2):S_3$), $C_2^3\times C_6$ ($(C_3^2\times C_6^2):S_3$)

Order 36: $C_6^2$ ($C_3^3:\GL(2,\mathbb{Z}/4)$) x 12, $C_6^2$ ($C_3^3:\GL(2,\mathbb{Z}/4)$) x 9, $C_6^2$ ($(C_3\times C_6^2):S_4$) x 4, $C_6^2$ ($(C_3\times C_6^2):S_4$) x 3

Order 32: $C_2^5$ ($C_3^5:D_6$)

Order 27: $C_3^3$ ($C_6^2:\GL(2,\mathbb{Z}/4)$) x 4, $C_3^3$ ($C_2^4.C_3^3:C_4.C_2$)

Order 24: $C_2^2\times C_6$ ($C_2\times C_6^2.C_3^3.C_2$) x 3, $C_2^2\times C_6$ ($C_2\times C_6^2.C_3^3.C_2$) x 3

Order 18: $C_3\times C_6$ ($C_6^3:S_4$) x 4, $C_3\times C_6$ ($C_6^3:S_4$) x 3

Order 16: $C_2^4$ ($C_3^5:S_4$) x 3, $C_2^4$ ($(C_3^3\times C_6^2):S_3$)

Order 12: $C_2\times C_6$ ($C_6^2.C_3^3.D_4$) x 3, $C_2\times C_6$ ($C_6^2.C_3^3.D_4$) x 3, $C_2\times C_6$ ($C_2^4.C_3\wr C_3.S_3$), $C_2\times C_6$ ($C_2^4.C_3^4:C_3.C_2$)

Order 9: $C_3^2$ ($(C_2\times C_6^3):S_4$) x 4, $C_3^2$ ($(C_2\times C_6^3):S_4$) x 3

Order 8: $C_2^3$ ($C_2\times C_3^5:S_4$) x 3

Order 6: $C_6$ ($C_2\times C_2^4.C_3\wr C_3.S_3$), $C_6$ ($C_2\times C_2^4.C_3^4:C_3.C_2$)

Order 4: $C_2^2$ ($C_3^5.\GL(2,\mathbb{Z}/4)$) x 3, $C_2^2$ ($(C_3^3\times C_6^2).S_4$)

Order 3: $C_3$ ($C_2^4.C_3^4:C_3.D_4$), $C_3$ ($C_2^4.C_3\wr C_3.C_3.D_4$)

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

Order 1: $C_1$ ($(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 93312: $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 46656: $C_2\times C_6^4.C_3.C_6$ ($C_2$), $(C_3^2\times C_6^3).S_4$ ($C_2$), $(C_3^2\times C_6^3).S_4$ ($C_2$)

Order 23328: $C_6^4.C_3.C_6$ ($C_2^2$)

Order 15552: $C_6\times (C_2\times C_6^3).C_6$ ($S_3$), $C_2\times C_2^4.C_3^4:C_3.C_2$ ($S_3$), $C_2\times C_6^5$ ($S_3$), $C_2\times C_2^4.C_3^2:C_9.C_6$ ($S_3$)

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

Order 7776: $C_6^5$ ($D_6$), $C_2^4.C_3^2:C_9.C_6$ ($D_6$), $C_3\times (C_2\times C_6^3).C_6$ ($D_6$), $C_2^4.C_3^4:C_3.C_2$ ($D_6$)

Order 5184: $C_2^2\times C_6^4$ ($C_3:S_3$) x 2, $C_6\times C_2^4:\He_3.C_2$ ($C_3:S_3$), $C_6\times C_2^4:C_9.C_6$ ($C_3:S_3$)

Order 3888: $C_3\times C_6^4$ ($C_3:D_4$), $C_3\times C_6^4$ ($S_4$), $C_2^4.C_3^4:C_3$ ($C_3:D_4$), $C_3\times C_2^4.C_3\wr C_3$ ($C_3:D_4$), $C_2^4.(C_3^4.C_3)$ ($C_3:D_4$)

Order 2592: $C_2\times C_6^4$ ($C_6:S_3$) x 2, $C_3\times C_2^4:\He_3.C_2$ ($C_6:S_3$), $C_3\times C_2^4:C_9.C_6$ ($C_6:S_3$)

Order 1944: $C_3^2\times C_6^3$ ($C_2\times S_4$)

Order 1728: $C_2^3\times C_6^3$ ($C_3^2:S_3$) x 2, $C_2^3\times C_6^3$ ($C_3^2:S_3$)

Order 1296: $C_6^4$ ($C_3:S_4$) x 2, $C_6^4$ ($C_6^2:C_2$) x 2, $C_3\times C_6^2:A_4$ ($C_6^2:C_2$), $C_3\times C_6^2.A_4$ ($C_6^2:C_2$)

Order 972: $C_3^3\times C_6^2$ ($C_2^2:S_4$), $C_3^3\times C_6^2$ ($\GL(2,\mathbb{Z}/4)$)

Order 864: $C_2^2\times C_6^3$ ($C_3^2:D_6$) x 2, $C_2^2\times C_6^3$ ($C_3^2:D_6$)

Order 648: $C_3\times C_6^3$ ($C_6:S_4$) x 2

Order 576: $C_2^4\times C_6^2$ ($C_3^3:S_3$) x 2, $C_2^4\times C_6^2$ ($C_3^3:S_3$)

Order 486: $C_3^4\times C_6$ ($C_2^3:S_4$)

Order 432: $C_2\times C_6^3$ ($C_3^2:S_4$) x 2, $C_2\times C_6^3$ ($C_6^2:S_3$) x 2, $C_2\times C_6^3$ ($C_3^2:S_4$), $C_2\times C_6^3$ ($C_6^2:S_3$)

Order 324: $C_3^2\times C_6^2$ ($C_3:\GL(2,\mathbb{Z}/4)$) x 2, $C_3^2\times C_6^2$ ($(C_2\times C_6):S_4$) x 2

Order 288: $C_2^3\times C_6^2$ ($C_3^3:D_6$) x 2, $C_2^3\times C_6^2$ ($C_3^3:D_6$)

Order 243: $C_3^5$ ($C_2\wr S_3$)

Order 216: $C_6^3$ ($C_6^2:D_6$) x 2, $C_6^3$ ($C_6^2:D_6$)

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

Order 162: $C_3^3\times C_6$ ($(C_2^2\times C_6):S_4$) x 2

Order 144: $C_2^2\times C_6^2$ ($C_3^3:S_4$) x 2, $C_2^2\times C_6^2$ ($(C_3\times C_6^2):S_3$) x 2, $C_2^2\times C_6^2$ ($C_3^3:S_4$), $C_2^2\times C_6^2$ ($(C_3\times C_6^2):S_3$)

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

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

Order 81: $C_3^4$ ($(C_2\times C_6):\GL(2,\mathbb{Z}/4)$) x 2

Order 72: $C_2\times C_6^2$ ($C_6^3:S_3$) x 2, $C_2\times C_6^2$ ($C_6^3:S_3$)

Order 64: $C_2^6$ ($C_3^5:S_3$)

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

Order 48: $C_2^3\times C_6$ ($C_3^4:S_4$), $C_2^3\times C_6$ ($C_3^4:S_4$), $C_2^3\times C_6$ ($(C_3^2\times C_6^2):S_3$), $C_2^3\times C_6$ ($(C_3^2\times C_6^2):S_3$)

Order 36: $C_6^2$ ($C_3^3:\GL(2,\mathbb{Z}/4)$) x 2, $C_6^2$ ($(C_3\times C_6^2):S_4$) x 2, $C_6^2$ ($(C_3\times C_6^2):S_4$), $C_6^2$ ($C_3^3:\GL(2,\mathbb{Z}/4)$)

Order 32: $C_2^5$ ($C_3^5:D_6$)

Order 27: $C_3^3$ ($C_6^2:\GL(2,\mathbb{Z}/4)$) x 2, $C_3^3$ ($C_2^4.C_3^3:C_4.C_2$)

Order 24: $C_2^2\times C_6$ ($C_2\times C_6^2.C_3^3.C_2$), $C_2^2\times C_6$ ($C_2\times C_6^2.C_3^3.C_2$)

Order 18: $C_3\times C_6$ ($C_6^3:S_4$) x 2, $C_3\times C_6$ ($C_6^3:S_4$)

Order 16: $C_2^4$ ($(C_3^3\times C_6^2):S_3$), $C_2^4$ ($C_3^5:S_4$)

Order 12: $C_2\times C_6$ ($C_2^4.C_3\wr C_3.S_3$), $C_2\times C_6$ ($C_6^2.C_3^3.D_4$), $C_2\times C_6$ ($C_2^4.C_3^4:C_3.C_2$), $C_2\times C_6$ ($C_6^2.C_3^3.D_4$)

Order 9: $C_3^2$ ($(C_2\times C_6^3):S_4$) x 2, $C_3^2$ ($(C_2\times C_6^3):S_4$)

Order 8: $C_2^3$ ($C_2\times C_3^5:S_4$)

Order 6: $C_6$ ($C_2\times C_2^4.C_3\wr C_3.S_3$), $C_6$ ($C_2\times C_2^4.C_3^4:C_3.C_2$)

Order 4: $C_2^2$ ($(C_3^3\times C_6^2).S_4$), $C_2^2$ ($C_3^5.\GL(2,\mathbb{Z}/4)$)

Order 3: $C_3$ ($C_2^4.C_3^4:C_3.D_4$), $C_3$ ($C_2^4.C_3\wr C_3.C_3.D_4$)

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

Order 1: $C_1$ ($(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$)

Series

Derived series

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

$\rhd$

$C_6^4.C_3.C_6$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$(C_3^2\times C_6^3).S_4$

$\rhd$

$C_6^4.C_3.C_6$

$\rhd$

$C_6^4.C_3^2$

$\rhd$

$C_3\times C_2^4.C_3\wr C_3$

$\rhd$

$C_6^4$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_6^4.C_3.C_6$

$\rhd$

$C_6^4.C_3^2$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_2\times C_6$

Supergroups

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

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

Character theory

Complex character table

The $2748 \times 2748$ character table is not available for this group.

Rational character table

The $1447 \times 1447$ rational character table is not available for this group.