Abstract group 5184.cn: $C_3\times C_6^3.D_4$

• Label: 5184.cn
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $20$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_3\times C_6^3.D_4$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^3.C_2^6.C_2^4$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	139	80	132	2048	432	1488	864	5184
Conjugacy classes	1	7	20	6	140	2	54	4	234
Divisions	1	7	12	6	77	2	30	2	137
Autjugacy classes	1	6	11	3	54	1	13	1	90

Dimension	1	2	4	8	16
Irr. complex chars.	24	42	120	48	0	234
Irr. rational chars.	8	18	32	58	21	137

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$1397760$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{6}=c^{6}=d^{4}=e^{6}=[c,e]=1, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,3,4,5,2,6)(7,8)(10,12)(11,14)(13,16)(15,17)(18,19,20), (1,3,4,6,2,5)(8,9)(10,11)(12,14)(13,16)(15,17), (1,2,4)(5,6)(11,13)(12,15)(16,17)(18,19,20)\rangle$
Transitive group:	24T7557				more information
Direct product:	$C_3$ $\, \times\, $ $(C_6^3.D_4)$
Semidirect product:	$(C_3^4:Q_8)$ $\,\rtimes\,$ $D_4$	$(C_6^2.D_6)$ $\,\rtimes\,$ $D_6$	$C_3^4$ $\,\rtimes\,$ $(D_4:D_4)$	$(C_6^3:D_6)$ $\,\rtimes\,$ $C_2$	all 31
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times C_6^3)$ . $D_4$	$(C_6^3:C_2)$ . $D_6$	$C_6^3$ . $(C_3:D_4)$	$C_6^3$ . $(C_3\times D_4)$	all 42

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

Homology

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

Subgroups

There are 23140 subgroups in 2309 conjugacy classes, 94 normal (62 characteristic).

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

Special subgroups

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

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 5184: $C_3\times C_6^3.D_4$

Order 2592: $C_6^2.\SOPlus(4,2)$ x 2, $C_6^2.\SOPlus(4,2)$ x 2, $C_3\times C_6^3.C_2^2$, $C_6^3.C_{12}$, $C_6^3:D_6$

Order 1728: $C_6^3.D_4$, $C_6^3.D_4$

Order 1296: $C_6^2.S_3^2$ x 4, $C_6^2:S_3^2$ x 3, $C_6^3:C_6$ x 2, $C_3^4:\OD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:D_8$ x 2, $C_6^3:C_6$, $C_6^2.C_6^2$, $C_6^2.C_6^2$, $C_3^2:C_{12}^2$

Order 864: $C_6^3.C_2^2$ x 3, $C_6^3:C_2^2$ x 2, $(C_3\times C_6^2).D_4$ x 2, $C_3\times C_6^2.D_4$ x 2, $(C_3\times C_6^2).D_4$ x 2, $C_3\times C_6^2.D_4$ x 2, $C_6^3:C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_4$, $C_6^3.C_4$

Order 648: $C_3^4:D_4$ x 6, $C_3^3:D_{12}$ x 4, $C_3\times C_6.S_3^2$ x 3, $C_3^4:C_2^3$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3^3:C_{24}$ x 2, $C_3\times C_6^3$, $C_3:C_6^3$, $C_3^4:C_2^3$, $C_6\times C_3^2:C_{12}$

Order 576: $C_2^3.\SOPlus(4,2)$, $C_{12}^2:C_2^2$

Order 432: $D_6:C_6^2$ x 16, $C_6^3:C_2$ x 6, $C_6^2:D_6$ x 6, $C_6^2.D_6$ x 6, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2:D_6$ x 3, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_3^3:C_4^2$ x 3, $C_3^3:\OD_{16}$ x 2, $C_6^2.C_{12}$ x 2, $C_3^3:\SD_{16}$ x 2, $C_3^3:D_8$ x 2, $C_3^3:\SD_{16}$ x 2, $C_3^3:D_8$ x 2, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_3:C_{12}^2$

Order 324: $C_3^2:C_6^2$ x 5, $C_3^3:C_{12}$ x 4, $C_3^2\times C_6^2$ x 3, $D_6\times C_3^3$ x 3, $C_3^3:C_{12}$ x 2, $C_3\wr C_2^2$

Order 288: $C_6^2.C_2^3$ x 4, $C_6^2.C_2^3$ x 2, $C_2^3:S_3^2$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_{12}\wr C_2$ x 2, $C_{12}^2:C_2$, $C_2^3.S_3^2$, $(C_2\times C_6^2).C_4$, $C_3\times C_{12}.D_4$

Order 216: $C_6^2:C_6$ x 48, $C_6^2:C_6$ x 18, $S_3\times C_6^2$ x 14, $C_6.C_6^2$ x 14, $C_3^2:D_{12}$ x 14, $C_6^3$ x 13, $C_3^3:D_4$ x 10, $C_6^2:C_6$ x 6, $C_6^2.C_6$ x 6, $C_6.S_3^2$ x 6, $C_3^2:D_{12}$ x 4, $C_3^3:Q_8$ x 4, $C_6\times S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $D_4\times C_3^3$ x 2, $C_3^2\times D_{12}$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:C_8$ x 2, $C_3^2:C_{24}$ x 2, $C_6:S_3^2$, $C_3\times C_6\times C_{12}$

Order 192: $C_3\times D_4:D_4$, $D_{12}:D_4$

Order 162: $C_3^3\times C_6$ x 3, $C_3^2\wr C_2$ x 3, $S_3\times C_3^3$

Order 144: $C_6^2:C_2^2$ x 37, $C_4:C_6^2$ x 13, $C_{12}.D_6$ x 6, $C_{12}:D_6$ x 6, $D_6:D_6$ x 6, $D_6.D_6$ x 6, $C_{12}:C_{12}$ x 4, $D_{12}:C_6$ x 4, $C_6\times D_{12}$ x 4, $D_6.D_6$ x 4, $D_6:D_6$ x 3, $C_{12}.D_6$ x 2, $D_{12}:C_6$ x 2, $C_{12}.C_{12}$ x 2, $C_6^2:C_2^2$ x 2, $C_3^2:\OD_{16}$ x 2, $C_3^2:\SD_{16}$ x 2, $C_3^2:D_8$ x 2, $C_3^2:C_4^2$, $C_6:D_{12}$, $D_6:D_6$, $C_{12}^2$

Order 108: $C_3\times C_6^2$ x 58, $C_3^2\times D_6$ x 47, $C_3^2:C_{12}$ x 28, $C_3^2:D_6$ x 17, $C_3^2:C_{12}$ x 12, $C_3\times S_3^2$ x 3, $C_3^2\times C_{12}$ x 2, $C_3:S_3^2$

Order 96: $C_{12}:D_4$ x 3, $D_4:C_{12}$ x 2, $D_4:D_6$ x 2, $D_8:C_6$ x 2, $D_{12}:C_2^2$ x 2, $D_{12}:C_4$ x 2, $\OD_{16}:C_6$, $C_{12}.D_4$, $C_6.C_2^4$, $C_{12}:D_4$

Order 81: $C_3^4$

Order 72: $C_6\wr C_2$ x 108, $C_2\times C_6^2$ x 41, $D_4\times C_3^2$ x 36, $C_6\times D_6$ x 29, $C_6:C_{12}$ x 29, $C_3\times D_{12}$ x 12, $C_6\times C_{12}$ x 11, $C_3:D_{12}$ x 8, $C_6^2:C_2$ x 6, $S_3\times C_{12}$ x 6, $D_6:S_3$ x 6, $C_6.D_6$ x 6, $C_3^2:Q_8$ x 4, $C_3^2:Q_8$ x 4, $S_3\times D_6$ x 3, $C_6.D_6$ x 3, $C_6:D_6$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$ x 2, $C_3:C_{24}$ x 2

Order 64: $D_4:D_4$

Order 54: $C_3^2\times C_6$ x 56, $S_3\times C_3^2$ x 22, $C_3^2:C_6$ x 9

Order 48: $C_6\times D_4$ x 25, $C_6:D_4$ x 11, $D_4:S_3$ x 6, $S_3\times D_4$ x 6, $D_4:C_6$ x 4, $D_{12}:C_2$ x 4, $C_4\times C_{12}$ x 3, $C_3\times \SD_{16}$ x 2, $C_3\times D_8$ x 2, $C_3\times \OD_{16}$ x 2, $C_3:\SD_{16}$ x 2, $C_3:D_8$ x 2, $C_3:\OD_{16}$ x 2, $C_2\times D_{12}$, $C_{12}:C_4$

Order 36: $C_6^2$ x 190, $C_6\times S_3$ x 92, $C_3:C_{12}$ x 52, $C_3\times C_{12}$ x 22, $C_6:S_3$ x 5, $C_3^2:C_4$ x 4, $S_3^2$ x 3

Order 32: $D_8:C_2$ x 2, $C_4\wr C_2$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $C_4:D_4$

Order 27: $C_3^3$ x 12

Order 24: $C_3\times D_4$ x 62, $C_3:D_4$ x 30, $C_2^2\times C_6$ x 28, $C_2\times C_{12}$ x 18, $C_6:C_4$ x 9, $C_2\times D_6$ x 9, $D_{12}$ x 6, $C_4\times S_3$ x 6, $C_3:Q_8$ x 4, $C_{24}$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2

Order 18: $C_3\times C_6$ x 170, $C_3\times S_3$ x 43, $C_3:S_3$ x 3

Order 16: $C_2\times D_4$ x 8, $D_4:C_2$ x 4, $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 104, $C_{12}$ x 30, $D_6$ x 22, $C_3:C_4$ x 12

Order 9: $C_3^2$ x 30

Order 8: $D_4$ x 16, $C_2\times C_4$ x 6, $C_2^3$ x 5, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 77, $S_3$ x 10

Order 4: $C_2^2$ x 12, $C_4$ x 6

Order 3: $C_3$ x 12

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $C_3\times C_6^3.D_4$

Order 2592: $C_3\times C_6^3.C_2^2$, $C_6^2.\SOPlus(4,2)$, $C_6^2.\SOPlus(4,2)$, $C_6^3.C_{12}$, $C_6^3:D_6$

Order 1728: $C_6^3.D_4$, $C_6^3.D_4$

Order 1296: $C_6^3:C_6$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3:C_6$, $C_6^2.C_6^2$, $C_6^2.C_6^2$, $C_3^4:\OD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:D_8$, $C_3^2:C_{12}^2$

Order 864: $C_6^3.C_2^2$ x 3, $C_6^3:C_2^2$ x 2, $C_6^3:C_2^2$, $(C_3\times C_6^2).D_4$, $C_3\times C_6^2.D_4$, $C_6^3.C_2^2$, $C_6^3.C_4$, $C_6^3.C_4$, $(C_3\times C_6^2).D_4$, $C_3\times C_6^2.D_4$

Order 648: $C_3^4:D_4$ x 3, $C_3^4:C_2^3$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3\times C_6.S_3^2$ x 2, $C_3\times C_6^3$, $C_3:C_6^3$, $C_3^4:C_2^3$, $C_3^4:D_4$, $C_6\times C_3^2:C_{12}$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^3:C_{24}$

Order 576: $C_2^3.\SOPlus(4,2)$, $C_{12}^2:C_2^2$

Order 432: $D_6:C_6^2$ x 14, $C_6^3:C_2$ x 6, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_3^3:C_4^2$ x 3, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_3^3:\OD_{16}$, $C_6^2.C_{12}$, $C_3^3:\SD_{16}$, $C_3^3:D_8$, $C_3^3:\SD_{16}$, $C_3^3:D_8$, $C_3:C_{12}^2$

Order 324: $C_3^2:C_6^2$ x 4, $C_3^2\times C_6^2$ x 3, $D_6\times C_3^3$ x 3, $C_3^3:C_{12}$ x 2, $C_3\wr C_2^2$, $C_3^3:C_{12}$

Order 288: $C_6^2.C_2^3$ x 3, $C_6^2.C_2^3$ x 2, $C_2^3:S_3^2$ x 2, $C_6^2.D_4$, $C_{12}^2:C_2$, $C_6^2.D_4$, $C_2^3.S_3^2$, $(C_2\times C_6^2).C_4$, $C_6^2.D_4$, $C_3\times C_{12}.D_4$, $C_{12}\wr C_2$

Order 216: $C_6^2:C_6$ x 18, $C_6^3$ x 12, $S_3\times C_6^2$ x 12, $C_6.C_6^2$ x 12, $C_6^2:C_6$ x 9, $C_6^2:C_6$ x 6, $C_6^2.C_6$ x 6, $C_3^2:D_{12}$ x 6, $C_3^3:D_4$ x 5, $C_6.S_3^2$ x 3, $C_6\times S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_3^3:Q_8$ x 2, $C_6.S_3^2$ x 2, $C_6:S_3^2$, $D_4\times C_3^3$, $C_3\times C_6\times C_{12}$, $C_3^2\times D_{12}$, $C_3^3:Q_8$, $C_3^3:C_8$, $C_3^2:C_{24}$

Order 192: $C_3\times D_4:D_4$, $D_{12}:D_4$

Order 162: $C_3^3\times C_6$ x 3, $C_3^2\wr C_2$ x 2, $S_3\times C_3^3$

Order 144: $C_6^2:C_2^2$ x 30, $C_4:C_6^2$ x 11, $C_{12}:C_{12}$ x 3, $C_{12}.D_6$ x 3, $C_{12}:D_6$ x 3, $C_6\times D_{12}$ x 3, $D_6:D_6$ x 3, $D_6.D_6$ x 3, $C_6^2:C_2^2$ x 2, $D_{12}:C_6$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_{12}.D_6$, $D_{12}:C_6$, $C_{12}.C_{12}$, $C_3^2:C_4^2$, $C_6:D_{12}$, $D_6:D_6$, $C_3^2:\OD_{16}$, $C_3^2:\SD_{16}$, $C_3^2:D_8$, $C_{12}^2$

Order 108: $C_3\times C_6^2$ x 44, $C_3^2\times D_6$ x 32, $C_3^2:D_6$ x 12, $C_3^2:C_{12}$ x 12, $C_3^2:C_{12}$ x 6, $C_3\times S_3^2$ x 2, $C_3:S_3^2$, $C_3^2\times C_{12}$

Order 96: $C_{12}:D_4$ x 3, $D_4:D_6$ x 2, $D_4:C_{12}$, $\OD_{16}:C_6$, $C_{12}.D_4$, $C_6.C_2^4$, $D_8:C_6$, $C_{12}:D_4$, $D_{12}:C_2^2$, $D_{12}:C_4$

Order 81: $C_3^4$

Order 72: $C_6\wr C_2$ x 43, $C_2\times C_6^2$ x 33, $C_6\times D_6$ x 24, $C_6:C_{12}$ x 24, $D_4\times C_3^2$ x 12, $C_6\times C_{12}$ x 9, $C_3\times D_{12}$ x 5, $C_3:D_{12}$ x 4, $C_6^2:C_2$ x 3, $S_3\times C_{12}$ x 3, $D_6:S_3$ x 3, $C_6.D_6$ x 3, $C_6:D_6$ x 2, $S_3\times D_6$ x 2, $C_6.D_6$ x 2, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $D_4:D_4$

Order 54: $C_3^2\times C_6$ x 42, $S_3\times C_3^2$ x 12, $C_3^2:C_6$ x 6

Order 48: $C_6\times D_4$ x 21, $C_6:D_4$ x 10, $D_4:S_3$ x 3, $S_3\times D_4$ x 3, $C_4\times C_{12}$ x 3, $D_4:C_6$ x 2, $D_{12}:C_2$ x 2, $C_2\times D_{12}$, $C_3\times \SD_{16}$, $C_3\times D_8$, $C_3\times \OD_{16}$, $C_3:\SD_{16}$, $C_3:D_8$, $C_{12}:C_4$, $C_3:\OD_{16}$

Order 36: $C_6^2$ x 121, $C_6\times S_3$ x 56, $C_3:C_{12}$ x 22, $C_3\times C_{12}$ x 9, $C_6:S_3$ x 4, $C_3^2:C_4$ x 2, $S_3^2$ x 2

Order 32: $\OD_{16}:C_2$, $D_4:C_2^2$, $D_8:C_2$, $C_4:D_4$, $C_4\wr C_2$

Order 27: $C_3^3$ x 11

Order 24: $C_2^2\times C_6$ x 25, $C_3\times D_4$ x 24, $C_2\times C_{12}$ x 15, $C_3:D_4$ x 15, $C_6:C_4$ x 8, $C_2\times D_6$ x 8, $D_{12}$ x 3, $C_4\times S_3$ x 3, $C_3:Q_8$ x 2, $C_{24}$, $C_3\times Q_8$, $C_3:C_8$

Order 18: $C_3\times C_6$ x 105, $C_3\times S_3$ x 22, $C_3:S_3$ x 2

Order 16: $C_2\times D_4$ x 7, $D_4:C_2$ x 2, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4^2$

Order 12: $C_2\times C_6$ x 75, $D_6$ x 16, $C_{12}$ x 13, $C_3:C_4$ x 6

Order 9: $C_3^2$ x 24

Order 8: $D_4$ x 8, $C_2^3$ x 5, $C_2\times C_4$ x 5, $Q_8$, $C_8$

Order 6: $C_6$ x 54, $S_3$ x 6

Order 4: $C_2^2$ x 11, $C_4$ x 3

Order 3: $C_3$ x 11

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $C_3\times C_6^3.D_4$ ($C_1$)

Order 2592: $C_6^2.\SOPlus(4,2)$ ($C_2$) x 2, $C_6^2.\SOPlus(4,2)$ ($C_2$) x 2, $C_3\times C_6^3.C_2^2$ ($C_2$), $C_6^3.C_{12}$ ($C_2$), $C_6^3:D_6$ ($C_2$)

Order 1728: $C_6^3.D_4$ ($C_3$)

Order 1296: $C_6^2.S_3^2$ ($C_2^2$) x 2, $C_3^4:\OD_{16}$ ($C_2^2$) x 2, $C_6^3:C_6$ ($C_2^2$), $C_6^2.C_6^2$ ($C_2^2$), $C_3^2:C_{12}^2$ ($C_2^2$)

Order 864: $(C_3\times C_6^2).D_4$ ($C_6$) x 2, $(C_3\times C_6^2).D_4$ ($C_6$) x 2, $C_6^3:C_2^2$ ($C_6$), $C_6^3.C_2^2$ ($C_6$), $C_6^3.C_2^2$ ($S_3$), $C_6^3.C_4$ ($C_6$)

Order 648: $C_3^4:Q_8$ ($D_4$) x 2, $C_3^3:D_{12}$ ($D_4$) x 2, $C_3\times C_6^3$ ($D_4$), $C_3^4:C_2^3$ ($D_4$), $C_6\times C_3^2:C_{12}$ ($C_2^3$)

Order 432: $C_6^2.D_6$ ($C_2\times C_6$) x 2, $C_3^3:\OD_{16}$ ($C_2\times C_6$) x 2, $C_6^3:C_2$ ($C_2\times C_6$), $C_6^3:C_2$ ($D_6$), $C_6^2.D_6$ ($C_2\times C_6$), $C_6^2.D_6$ ($D_6$), $C_3^3:C_4^2$ ($C_2\times C_6$), $C_3^3:C_4^2$ ($D_6$)

Order 324: $C_3^3:C_{12}$ ($C_2\times D_4$) x 2, $C_3^2\times C_6^2$ ($C_2\times D_4$)

Order 288: $C_2^3.S_3^2$ ($C_3\times S_3$)

Order 216: $C_3^3:Q_8$ ($C_3\times D_4$) x 2, $C_3^2:D_{12}$ ($C_3\times D_4$) x 2, $C_6^3$ ($C_3:D_4$), $C_6^3$ ($C_3\times D_4$), $C_6^2:C_6$ ($C_3:D_4$), $C_6^2:C_6$ ($C_3\times D_4$), $C_6^2.C_6$ ($C_2^2\times C_6$), $C_6^2.C_6$ ($C_2\times D_6$)

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

Order 144: $C_3^2:C_4^2$ ($C_6\times S_3$), $C_6^2:C_2^2$ ($C_6\times S_3$), $D_6:D_6$ ($C_6\times S_3$)

Order 108: $C_3^2:C_{12}$ ($C_6\times D_4$) x 2, $C_3^2:C_{12}$ ($S_3\times D_4$) x 2, $C_3\times C_6^2$ ($C_6\times D_4$), $C_3\times C_6^2$ ($C_6:D_4$)

Order 81: $C_3^4$ ($D_4:D_4$)

Order 72: $C_2\times C_6^2$ ($\SOPlus(4,2)$), $C_2\times C_6^2$ ($C_6\wr C_2$), $C_6:D_6$ ($C_6\wr C_2$), $C_6.D_6$ ($C_6\times D_6$)

Order 54: $C_3^2\times C_6$ ($C_2^4:C_6$), $C_3^2\times C_6$ ($D_6:D_4$)

Order 36: $C_3^2:C_4$ ($C_{12}:D_6$) x 2, $C_6^2$ ($S_3^2:C_2^2$), $C_6^2$ ($C_6^2:C_2^2$)

Order 27: $C_3^3$ ($C_3\times D_4:D_4$), $C_3^3$ ($D_{12}:D_4$)

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

Order 18: $C_3\times C_6$ ($D_6\wr C_2$), $C_3\times C_6$ ($C_6^2:D_4$)

Order 12: $C_2\times C_6$ ($S_3^2:D_6$), $C_2\times C_6$ ($C_6\times \SOPlus(4,2)$)

Order 9: $C_3^2$ ($C_2^3.\SOPlus(4,2)$), $C_3^2$ ($C_{12}^2:C_2^2$)

Order 8: $C_2^3$ ($C_3\wr D_4$)

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

Order 4: $C_2^2$ ($C_2\times C_3\wr D_4$)

Order 3: $C_3$ ($C_6^3.D_4$), $C_3$ ($C_6^3.D_4$)

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

Order 1: $C_1$ ($C_3\times C_6^3.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $C_3\times C_6^3.D_4$ ($C_1$)

Order 2592: $C_3\times C_6^3.C_2^2$ ($C_2$), $C_6^2.\SOPlus(4,2)$ ($C_2$), $C_6^2.\SOPlus(4,2)$ ($C_2$), $C_6^3.C_{12}$ ($C_2$), $C_6^3:D_6$ ($C_2$)

Order 1728: $C_6^3.D_4$ ($C_3$)

Order 1296: $C_6^3:C_6$ ($C_2^2$), $C_6^2.S_3^2$ ($C_2^2$), $C_6^2.C_6^2$ ($C_2^2$), $C_3^4:\OD_{16}$ ($C_2^2$), $C_3^2:C_{12}^2$ ($C_2^2$)

Order 864: $C_6^3:C_2^2$ ($C_6$), $(C_3\times C_6^2).D_4$ ($C_6$), $C_6^3.C_2^2$ ($C_6$), $C_6^3.C_2^2$ ($S_3$), $C_6^3.C_4$ ($C_6$), $(C_3\times C_6^2).D_4$ ($C_6$)

Order 648: $C_3\times C_6^3$ ($D_4$), $C_3^4:C_2^3$ ($D_4$), $C_6\times C_3^2:C_{12}$ ($C_2^3$), $C_3^4:Q_8$ ($D_4$), $C_3^3:D_{12}$ ($D_4$)

Order 432: $C_6^3:C_2$ ($C_2\times C_6$), $C_6^3:C_2$ ($D_6$), $C_6^2.D_6$ ($C_2\times C_6$), $C_6^2.D_6$ ($C_2\times C_6$), $C_6^2.D_6$ ($D_6$), $C_3^3:\OD_{16}$ ($C_2\times C_6$), $C_3^3:C_4^2$ ($C_2\times C_6$), $C_3^3:C_4^2$ ($D_6$)

Order 324: $C_3^2\times C_6^2$ ($C_2\times D_4$), $C_3^3:C_{12}$ ($C_2\times D_4$)

Order 288: $C_2^3.S_3^2$ ($C_3\times S_3$)

Order 216: $C_6^3$ ($C_3:D_4$), $C_6^3$ ($C_3\times D_4$), $C_6^2:C_6$ ($C_3:D_4$), $C_6^2:C_6$ ($C_3\times D_4$), $C_6^2.C_6$ ($C_2^2\times C_6$), $C_6^2.C_6$ ($C_2\times D_6$), $C_3^3:Q_8$ ($C_3\times D_4$), $C_3^2:D_{12}$ ($C_3\times D_4$)

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

Order 144: $C_3^2:C_4^2$ ($C_6\times S_3$), $C_6^2:C_2^2$ ($C_6\times S_3$), $D_6:D_6$ ($C_6\times S_3$)

Order 108: $C_3\times C_6^2$ ($C_6\times D_4$), $C_3\times C_6^2$ ($C_6:D_4$), $C_3^2:C_{12}$ ($C_6\times D_4$), $C_3^2:C_{12}$ ($S_3\times D_4$)

Order 81: $C_3^4$ ($D_4:D_4$)

Order 72: $C_2\times C_6^2$ ($\SOPlus(4,2)$), $C_2\times C_6^2$ ($C_6\wr C_2$), $C_6:D_6$ ($C_6\wr C_2$), $C_6.D_6$ ($C_6\times D_6$)

Order 54: $C_3^2\times C_6$ ($C_2^4:C_6$), $C_3^2\times C_6$ ($D_6:D_4$)

Order 36: $C_3^2:C_4$ ($C_{12}:D_6$), $C_6^2$ ($S_3^2:C_2^2$), $C_6^2$ ($C_6^2:C_2^2$)

Order 27: $C_3^3$ ($C_3\times D_4:D_4$), $C_3^3$ ($D_{12}:D_4$)

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

Order 18: $C_3\times C_6$ ($D_6\wr C_2$), $C_3\times C_6$ ($C_6^2:D_4$)

Order 12: $C_2\times C_6$ ($S_3^2:D_6$), $C_2\times C_6$ ($C_6\times \SOPlus(4,2)$)

Order 9: $C_3^2$ ($C_2^3.\SOPlus(4,2)$), $C_3^2$ ($C_{12}^2:C_2^2$)

Order 8: $C_2^3$ ($C_3\wr D_4$)

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

Order 4: $C_2^2$ ($C_2\times C_3\wr D_4$)

Order 3: $C_3$ ($C_6^3.D_4$), $C_3$ ($C_6^3.D_4$)

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

Order 1: $C_1$ ($C_3\times C_6^3.D_4$)

Series

Derived series

$C_3\times C_6^3.D_4$

$\rhd$

$C_3\times C_6^3.D_4$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3\times C_6^3.D_4$

$\rhd$

$C_3\times C_6^3.D_4$

$\rhd$

$C_6^3:D_6$

$\rhd$

$C_6^3:D_6$

$\rhd$

$C_6^2.S_3^2$

$\rhd$

$C_6^2.S_3^2$

$\rhd$

$C_6\times C_3^2:C_{12}$

$\rhd$

$C_6\times C_3^2:C_{12}$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3\times C_6^3.D_4$

$\rhd$

$C_3\times C_6^3.D_4$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_6$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_2^2\times C_6$

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

See the $234 \times 234$ 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 $137 \times 137$ rational character table (warning: may be slow to load).