Abstract group 7776.ea: $C_3^4.\GL(2,\mathbb{Z}/4)$

• Label: 7776.ea
• Order: \( 2^{5} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $26$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^4.\GL(2,\mathbb{Z}/4)$
Order:	\(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2\times C_6^2.C_3^3.C_2^3$, of order \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \)
Composition factors:	$C_2$ x 5, $C_3$ x 5
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	187	80	324	2000	648	2592	1944	7776
Conjugacy classes	1	6	13	3	85	6	24	12	150
Divisions	1	6	9	3	51	4	10	7	91
Autjugacy classes	1	6	13	3	78	6	15	9	131

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	12	27	12	6	55	0	38	0	150
Irr. rational chars.	4	9	4	13	15	2	28	16	91

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$432$

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:	${\langle a, b, c, d, e \mid a^{6}=b^{18}=c^{6}=d^{2}=e^{6}=[a,d]=[b,d]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(1,2)(3,5)(4,6)(7,8)(10,11,12,13)(14,16,20,22,19,17)(15,18,21)(23,24,26,25), (2,3,4,5,7,9)(6,8)(11,12)(14,15,17)(16,19,21,22,20,18)(24,25)\rangle$
Transitive group:	36T7108				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^3.S_3)$ $\,\rtimes\,$ $S_3$	$(C_6^3.S_3)$ $\,\rtimes\,$ $C_6$	$(C_3^4.A_4)$ $\,\rtimes\,$ $D_4$	$(C_6^3.C_6)$ $\,\rtimes\,$ $C_6$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^3$ . $S_3^2$	$C_3^3$ . $(D_6:S_4)$	$(C_6^3:C_6)$ . $S_3$	$C_6^3$ . $(C_6\times S_3)$	all 39

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

Homology

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

Subgroups

There are 27722 subgroups in 1650 conjugacy classes, 60 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

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 7776: $C_3^4.\GL(2,\mathbb{Z}/4)$

Order 3888: $C_2\times C_6^2.C_3^3.C_2$, $(C_3\times C_6^3).S_3$, $(C_3\times C_6^3).C_6$

Order 2592: $C_3^3.(C_6\times D_4).C_2$, $C_3^3.\GL(2,\mathbb{Z}/4)$, $C_3^3.\GL(2,\mathbb{Z}/4)$

Order 1944: $C_6^3.C_3^2$, $C_3^3.(C_6\times A_4)$, $C_3^4.S_4$, $(C_3^3\times C_6).D_6$

Order 1296: $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.S_3$ x 2, $C_6^3.C_6$ x 2, $C_6^2:D_{18}$ x 2, $C_6^3.S_3$, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^3.C_6$, $C_6^3.S_3$, $C_6^3.C_6$

Order 972: $C_3^4.D_6$, $C_3^3.C_6^2$, $C_3^4.A_4$, $(C_3^3\times C_6).S_3$

Order 864: $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^3.C_2^2$, $C_6^2.S_4$, $C_3^2.\GL(2,\mathbb{Z}/4)$

Order 648: $C_3^4:C_2^3$ x 5, $C_3^3:D_{12}$ x 4, $C_6^2:C_{18}$ x 4, $C_6\times C_3^2:C_{12}$ x 3, $C_6^2:C_{18}$ x 3, $C_6^2:D_9$ x 3, $C_3^4:D_4$ x 2, $C_6^3.C_3$ x 2, $C_3\times C_6^3$, $S_3\times C_3^2.A_4$, $C_3^3.S_4$, $(C_3\times C_6).D_{18}$, $(S_3\times C_{18}):C_6$

Order 486: $C_3^4.C_6$, $C_3^4.S_3$, $C_3^4.C_6$

Order 432: $(C_3\times C_6).S_4$ x 5, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^3:C_2$ x 3, $D_6:C_6^2$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2:C_{12}$ x 3, $C_6^3:C_2$ x 2, $D_6\times C_6^2$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_2^2:C_9\times D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_3\times C_6.S_4$ x 2, $C_6^2.A_4$, $C_6^2.D_6$, $C_6^2.D_6$

Order 324: $C_3^2:C_6^2$ x 10, $C_6^2:C_9$ x 4, $C_3^2\times C_6^2$ x 3, $C_3^3:C_{12}$ x 3, $C_3\times C_9:C_{12}$ x 2, $C_3^2:D_{18}$ x 2, $C_2\times C_3^2:C_{18}$ x 2, $C_3^3.A_4$ x 2, $(C_3\times C_{18}).S_3$, $C_3^3.D_6$, $C_3^2.C_6^2$

Order 288: $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_2^4:D_9$

Order 243: $C_3^4.C_3$

Order 216: $C_6^2:C_6$ x 15, $S_3\times C_6^2$ x 15, $C_6^3$ x 9, $C_6^2.C_6$ x 9, $C_6.C_6^2$ x 9, $C_6^2:C_6$ x 8, $C_6^2:C_6$ x 8, $C_3^2:D_{12}$ x 8, $C_6^2.C_6$ x 6, $C_3^2:D_{12}$ x 4, $C_3^3:D_4$ x 4, $C_6^2.C_6$ x 4, $C_6^2.C_6$ x 3, $C_3^2.S_4$, $C_3^2.S_4$, $D_{18}:C_6$, $C_9:D_{12}$

Order 162: $C_3^2:C_{18}$ x 4, $C_3^3\times C_6$ x 3, $C_3^2\wr C_2$ x 3, $C_3^2:D_9$ x 3, $C_3^2:C_{18}$ x 3, $C_{18}:C_3^2$ x 2, $C_3^3.S_3$, $C_3^3.C_6$

Order 144: $C_6^2:C_2^2$ x 14, $C_6^2:C_4$ x 12, $C_6^2:C_2^2$ x 5, $D_6:C_{12}$ x 4, $C_6.D_{12}$ x 4, $C_6:D_{12}$ x 4, $C_6.D_{12}$ x 2, $C_6.S_4$ x 2, $C_6\times D_{12}$ x 2, $D_6:D_6$ x 2, $C_2^3:C_{18}$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_2^2:D_{18}$, $C_6^2:C_4$, $C_6^2:C_4$

Order 108: $C_3\times C_6^2$ x 41, $C_3^2:D_6$ x 33, $C_3^2\times D_6$ x 33, $C_3^2:C_{12}$ x 10, $C_3^2:C_{12}$ x 10, $C_9:C_{12}$ x 5, $C_3^2.A_4$ x 5, $C_3^2.A_4$ x 4, $C_9:C_{12}$ x 2, $S_3\times C_{18}$ x 2, $C_{18}:C_6$, $C_3:D_{18}$, $C_{18}:C_6$

Order 96: $C_2^2:D_{12}$, $C_2^4:C_6$, $C_2^4:S_3$, $D_6:D_4$

Order 81: $C_3^2:C_9$ x 4, $C_9:C_3^2$ x 2, $C_3^4$

Order 72: $C_6\times D_6$ x 52, $C_6\wr C_2$ x 40, $C_6:C_{12}$ x 30, $C_2\times C_6^2$ x 29, $C_3:D_{12}$ x 8, $C_2^2:C_{18}$ x 7, $C_6:D_6$ x 5, $C_3\times D_{12}$ x 4, $D_6:S_3$ x 4, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3, $D_4\times C_3^2$ x 2, $C_6^2:C_2$ x 2, $C_9:D_4$, $C_2^2:D_9$

Order 54: $C_3^2\times C_6$ x 41, $C_3^2:C_6$ x 10, $S_3\times C_3^2$ x 10, $C_9:C_6$ x 6, $C_3\times C_{18}$ x 4, $S_3\times C_9$ x 3, $C_3:D_9$, $C_9:C_6$

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

Order 36: $C_6^2$ x 126, $C_6\times S_3$ x 119, $C_3:C_{12}$ x 36, $C_6:S_3$ x 10, $C_2^2:C_9$ x 4, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3, $C_2\times C_{18}$ x 2, $C_9:C_4$ x 2, $D_{18}$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 9, $C_9:C_3$ x 5, $C_3\times C_9$ x 4

Order 24: $C_2^2\times C_6$ x 24, $C_3:D_4$ x 16, $C_2\times D_6$ x 15, $C_3\times D_4$ x 12, $C_2\times C_{12}$ x 9, $C_6:C_4$ x 9, $D_{12}$ x 4

Order 18: $C_3\times C_6$ x 119, $C_3\times S_3$ x 38, $C_{18}$ x 7, $C_3:S_3$ x 3, $D_9$

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

Order 12: $C_2\times C_6$ x 74, $D_6$ x 33, $C_{12}$ x 10, $C_3:C_4$ x 10

Order 9: $C_3^2$ x 24, $C_9$ x 4

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

Order 6: $C_6$ x 51, $S_3$ x 10

Order 4: $C_2^2$ x 13, $C_4$ x 3

Order 3: $C_3$ x 9

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 7776: $C_3^4.\GL(2,\mathbb{Z}/4)$

Order 3888: $C_2\times C_6^2.C_3^3.C_2$, $(C_3\times C_6^3).S_3$, $(C_3\times C_6^3).C_6$

Order 2592: $C_3^3.(C_6\times D_4).C_2$, $C_3^3.\GL(2,\mathbb{Z}/4)$, $C_3^3.\GL(2,\mathbb{Z}/4)$

Order 1944: $C_6^3.C_3^2$, $C_3^3.(C_6\times A_4)$, $C_3^4.S_4$, $(C_3^3\times C_6).D_6$

Order 1296: $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.S_3$ x 2, $C_6^3.C_6$ x 2, $C_6^2:D_{18}$ x 2, $C_6^3.S_3$, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^3.C_6$, $C_6^3.S_3$, $C_6^3.C_6$

Order 972: $C_3^4.D_6$, $C_3^3.C_6^2$, $C_3^4.A_4$, $(C_3^3\times C_6).S_3$

Order 864: $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^3.C_2^2$, $C_6^2.S_4$, $C_3^2.\GL(2,\mathbb{Z}/4)$

Order 648: $C_3^4:C_2^3$ x 5, $C_6^2:C_{18}$ x 4, $C_6\times C_3^2:C_{12}$ x 3, $C_3^3:D_{12}$ x 3, $C_3^4:D_4$ x 2, $C_6^3.C_3$ x 2, $C_6^2:C_{18}$ x 2, $C_6^2:D_9$ x 2, $C_3\times C_6^3$, $S_3\times C_3^2.A_4$, $C_3^3.S_4$, $(C_3\times C_6).D_{18}$, $(S_3\times C_{18}):C_6$

Order 486: $C_3^4.C_6$, $C_3^4.S_3$, $C_3^4.C_6$

Order 432: $(C_3\times C_6).S_4$ x 5, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^3:C_2$ x 3, $D_6:C_6^2$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2:C_{12}$ x 3, $C_6^3:C_2$ x 2, $D_6\times C_6^2$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_2^2:C_9\times D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_3\times C_6.S_4$ x 2, $C_6^2.A_4$, $C_6^2.D_6$, $C_6^2.D_6$

Order 324: $C_3^2:C_6^2$ x 9, $C_6^2:C_9$ x 4, $C_3^2\times C_6^2$ x 3, $C_3^3:C_{12}$ x 3, $C_3\times C_9:C_{12}$ x 2, $C_3^2:D_{18}$ x 2, $C_2\times C_3^2:C_{18}$ x 2, $C_3^3.A_4$ x 2, $(C_3\times C_{18}).S_3$, $C_3^3.D_6$, $C_3^2.C_6^2$

Order 288: $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_2^4:D_9$

Order 243: $C_3^4.C_3$

Order 216: $C_6^2:C_6$ x 14, $S_3\times C_6^2$ x 14, $C_6^3$ x 9, $C_6^2.C_6$ x 9, $C_6.C_6^2$ x 9, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 6, $C_3^2:D_{12}$ x 6, $C_6^2.C_6$ x 6, $C_6^2.C_6$ x 4, $C_3^2:D_{12}$ x 3, $C_3^3:D_4$ x 3, $C_6^2.C_6$ x 2, $C_3^2.S_4$, $C_3^2.S_4$, $D_{18}:C_6$, $C_9:D_{12}$

Order 162: $C_3^2:C_{18}$ x 4, $C_3^3\times C_6$ x 3, $C_3^2\wr C_2$ x 3, $C_3^2:D_9$ x 2, $C_{18}:C_3^2$ x 2, $C_3^2:C_{18}$ x 2, $C_3^3.S_3$, $C_3^3.C_6$

Order 144: $C_6^2:C_2^2$ x 14, $C_6^2:C_4$ x 12, $C_6^2:C_2^2$ x 5, $D_6:C_{12}$ x 4, $C_6.D_{12}$ x 4, $C_6:D_{12}$ x 4, $C_6.D_{12}$ x 2, $C_6.S_4$ x 2, $C_6\times D_{12}$ x 2, $D_6:D_6$ x 2, $C_2^3:C_{18}$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_2^2:D_{18}$, $C_6^2:C_4$, $C_6^2:C_4$

Order 108: $C_3\times C_6^2$ x 41, $C_3^2:D_6$ x 26, $C_3^2\times D_6$ x 26, $C_3^2:C_{12}$ x 9, $C_3^2:C_{12}$ x 9, $C_9:C_{12}$ x 5, $C_3^2.A_4$ x 5, $C_3^2.A_4$ x 4, $C_9:C_{12}$ x 2, $S_3\times C_{18}$ x 2, $C_{18}:C_6$, $C_3:D_{18}$, $C_{18}:C_6$

Order 96: $C_2^2:D_{12}$, $C_2^4:C_6$, $C_2^4:S_3$, $D_6:D_4$

Order 81: $C_3^2:C_9$ x 4, $C_9:C_3^2$ x 2, $C_3^4$

Order 72: $C_6\times D_6$ x 45, $C_6:C_{12}$ x 30, $C_2\times C_6^2$ x 29, $C_6\wr C_2$ x 26, $C_3:D_{12}$ x 6, $C_2^2:C_{18}$ x 6, $C_6:D_6$ x 5, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_3\times D_{12}$ x 3, $D_6:S_3$ x 3, $D_4\times C_3^2$ x 2, $C_6^2:C_2$ x 2, $C_9:D_4$, $C_2^2:D_9$

Order 54: $C_3^2\times C_6$ x 41, $C_3^2:C_6$ x 8, $S_3\times C_3^2$ x 8, $C_9:C_6$ x 6, $C_3\times C_{18}$ x 4, $S_3\times C_9$ x 2, $C_3:D_9$, $C_9:C_6$

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

Order 36: $C_6^2$ x 125, $C_6\times S_3$ x 85, $C_3:C_{12}$ x 30, $C_6:S_3$ x 9, $C_2^2:C_9$ x 4, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3, $C_2\times C_{18}$ x 2, $C_9:C_4$ x 2, $D_{18}$

Order 32: $C_2^2\wr C_2$

Order 27: $C_3^3$ x 9, $C_9:C_3$ x 5, $C_3\times C_9$ x 4

Order 24: $C_2^2\times C_6$ x 23, $C_2\times D_6$ x 14, $C_3:D_4$ x 12, $C_2\times C_{12}$ x 9, $C_6:C_4$ x 9, $C_3\times D_4$ x 9, $D_{12}$ x 3

Order 18: $C_3\times C_6$ x 119, $C_3\times S_3$ x 25, $C_{18}$ x 6, $C_3:S_3$ x 3, $D_9$

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

Order 12: $C_2\times C_6$ x 67, $D_6$ x 26, $C_{12}$ x 9, $C_3:C_4$ x 9

Order 9: $C_3^2$ x 24, $C_9$ x 4

Order 8: $C_2^3$ x 6, $D_4$ x 5, $C_2\times C_4$ x 3

Order 6: $C_6$ x 49, $S_3$ x 8

Order 4: $C_2^2$ x 12, $C_4$ x 3

Order 3: $C_3$ x 9

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7776: $C_3^4.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 3888: $C_2\times C_6^2.C_3^3.C_2$ ($C_2$), $(C_3\times C_6^3).S_3$ ($C_2$), $(C_3\times C_6^3).C_6$ ($C_2$)

Order 2592: $C_3^3.\GL(2,\mathbb{Z}/4)$ ($C_3$)

Order 1944: $C_6^3.C_3^2$ ($C_2^2$)

Order 1296: $C_6^3.S_3$ ($C_6$), $C_6^3:C_6$ ($S_3$), $C_6^3.S_3$ ($S_3$), $C_6^3.C_6$ ($C_6$), $C_6^2:D_{18}$ ($C_6$)

Order 972: $C_3^4.A_4$ ($D_4$)

Order 648: $C_3\times C_6^3$ ($D_6$), $C_6^3.C_3$ ($D_6$), $C_6^2:C_{18}$ ($C_2\times C_6$)

Order 432: $C_6^3:C_2$ ($C_3\times S_3$), $C_6^2.D_6$ ($C_3\times S_3$)

Order 324: $C_6^2:C_9$ ($C_3\times D_4$), $C_3^2\times C_6^2$ ($C_3:D_4$), $C_3^2:C_6^2$ ($S_4$), $C_3^3.A_4$ ($C_3:D_4$)

Order 216: $C_6^3$ ($C_6\times S_3$), $C_6^3$ ($S_3^2$), $C_6^2.C_6$ ($C_6\times S_3$)

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

Order 144: $C_6^2:C_2^2$ ($C_9:C_6$)

Order 108: $C_3\times C_6^2$ ($C_6\wr C_2$), $C_3\times C_6^2$ ($D_6:S_3$), $C_3^2:D_6$ ($C_3\times S_4$), $C_3^2.A_4$ ($C_6\wr C_2$)

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

Order 72: $C_2\times C_6^2$ ($C_3\times S_3^2$), $C_2\times C_6^2$ ($C_{18}:C_6$), $C_2\times C_6^2$ ($C_3^2:D_6$)

Order 54: $C_3^2\times C_6$ ($C_6\times S_4$), $C_3^2\times C_6$ ($S_3\times S_4$)

Order 36: $C_6^2$ ($D_{18}:C_6$), $C_6^2$ ($\He_3:D_4$), $C_6^2$ ($C_3^3:D_4$), $C_6:S_3$ ($C_3^2.S_4$)

Order 27: $C_3^3$ ($C_3\times \GL(2,\mathbb{Z}/4)$), $C_3^3$ ($D_6:S_4$)

Order 24: $C_2^2\times C_6$ ($C_3^2.S_3^2$), $C_2^2\times C_6$ ($C_3^3:D_6$)

Order 18: $C_3\times C_6$ ($C_6^2:D_6$), $C_3\times C_6$ ($C_6^2.D_6$), $C_3\times C_6$ ($C_6^2:D_6$)

Order 12: $C_2\times C_6$ ($(S_3\times C_{18}):C_6$), $C_2\times C_6$ ($C_3\times \He_3:D_4$)

Order 9: $C_3^2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$), $C_3^2$ ($C_6^2.S_4$), $C_3^2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_3^3.S_3^2$)

Order 6: $C_6$ ($C_6^2.S_3^2$), $C_6$ ($C_6^2.S_3^2$)

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

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 7776: $C_3^4.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 3888: $C_2\times C_6^2.C_3^3.C_2$ ($C_2$), $(C_3\times C_6^3).S_3$ ($C_2$), $(C_3\times C_6^3).C_6$ ($C_2$)

Order 2592: $C_3^3.\GL(2,\mathbb{Z}/4)$ ($C_3$)

Order 1944: $C_6^3.C_3^2$ ($C_2^2$)

Order 1296: $C_6^3.S_3$ ($C_6$), $C_6^3:C_6$ ($S_3$), $C_6^3.S_3$ ($S_3$), $C_6^3.C_6$ ($C_6$), $C_6^2:D_{18}$ ($C_6$)

Order 972: $C_3^4.A_4$ ($D_4$)

Order 648: $C_3\times C_6^3$ ($D_6$), $C_6^3.C_3$ ($D_6$), $C_6^2:C_{18}$ ($C_2\times C_6$)

Order 432: $C_6^3:C_2$ ($C_3\times S_3$), $C_6^2.D_6$ ($C_3\times S_3$)

Order 324: $C_6^2:C_9$ ($C_3\times D_4$), $C_3^2\times C_6^2$ ($C_3:D_4$), $C_3^2:C_6^2$ ($S_4$), $C_3^3.A_4$ ($C_3:D_4$)

Order 216: $C_6^3$ ($C_6\times S_3$), $C_6^3$ ($S_3^2$), $C_6^2.C_6$ ($C_6\times S_3$)

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

Order 144: $C_6^2:C_2^2$ ($C_9:C_6$)

Order 108: $C_3\times C_6^2$ ($C_6\wr C_2$), $C_3\times C_6^2$ ($D_6:S_3$), $C_3^2:D_6$ ($C_3\times S_4$), $C_3^2.A_4$ ($C_6\wr C_2$)

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

Order 72: $C_2\times C_6^2$ ($C_3\times S_3^2$), $C_2\times C_6^2$ ($C_{18}:C_6$), $C_2\times C_6^2$ ($C_3^2:D_6$)

Order 54: $C_3^2\times C_6$ ($C_6\times S_4$), $C_3^2\times C_6$ ($S_3\times S_4$)

Order 36: $C_6^2$ ($D_{18}:C_6$), $C_6^2$ ($\He_3:D_4$), $C_6^2$ ($C_3^3:D_4$), $C_6:S_3$ ($C_3^2.S_4$)

Order 27: $C_3^3$ ($C_3\times \GL(2,\mathbb{Z}/4)$), $C_3^3$ ($D_6:S_4$)

Order 24: $C_2^2\times C_6$ ($C_3^2.S_3^2$), $C_2^2\times C_6$ ($C_3^3:D_6$)

Order 18: $C_3\times C_6$ ($C_6^2:D_6$), $C_3\times C_6$ ($C_6^2.D_6$), $C_3\times C_6$ ($C_6^2:D_6$)

Order 12: $C_2\times C_6$ ($(S_3\times C_{18}):C_6$), $C_2\times C_6$ ($C_3\times \He_3:D_4$)

Order 9: $C_3^2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$), $C_3^2$ ($C_6^2.S_4$), $C_3^2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_3^3.S_3^2$)

Order 6: $C_6$ ($C_6^2.S_3^2$), $C_6$ ($C_6^2.S_3^2$)

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

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

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

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

Series

Derived series

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

$\rhd$

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

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

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

$\rhd$

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

$\rhd$

$C_2\times C_6^2.C_3^3.C_2$

$\rhd$

$C_2\times C_6^2.C_3^3.C_2$

$\rhd$

$C_6^3.C_3^2$

$\rhd$

$C_6^3.C_3^2$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

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

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_6^2:C_9$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $150 \times 150$ 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 $91 \times 91$ rational character table.