Abstract group 5184.in: $C_6^2:(D_4\times D_9)$

• Label: 5184.in
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^2:(D_4\times D_9)$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2^2\times C_6^2.C_3^3.C_2^3$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
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	9	12	18	36
Elements	1	411	26	612	1182	216	1224	1080	432	5184
Conjugacy classes	1	11	5	8	50	6	9	18	3	111
Divisions	1	11	5	8	40	2	8	5	1	81
Autjugacy classes	1	10	5	7	39	2	7	5	1	77

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	8	26	8	20	22	0	27	0	111
Irr. rational chars.	8	10	8	5	22	1	20	7	81

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$1088640$

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:	${\langle a, b, c, d, e \mid b^{18}=c^{4}=d^{6}=e^{6}=[a,e]=[d,e]=1, a^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(2,5)(4,9)(6,11)(7,13)(8,10)(12,15)(14,17)(16,18)(19,21,20,22)(23,24)(25,26) \!\cdots\! \rangle$
Transitive group:	36T5635				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^2.S_4)$ $\,\rtimes\,$ $S_3$	$(C_6^2.A_4)$ $\,\rtimes\,$ $D_6$	$(C_6^2:D_9)$ $\,\rtimes\,$ $D_4$	$(C_6^2.D_6)$ $\,\rtimes\,$ $D_6$	all 32
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $(S_3\times S_4)$	$C_3^3$ . $(D_4\times S_4)$	$(C_6^2:C_6)$ . $S_4$	$(C_6^2.D_6)$ . $D_6$	all 38

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

Homology

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

Subgroups

There are 44747 subgroups in 2030 conjugacy classes, 75 normal (71 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_6^3.D_6$
Commutator:	$G' \simeq$ $C_6^2:C_{18}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3\times C_6$	$G/\Phi \simeq$ $D_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6^2:(D_4\times D_9)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $S_3\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2:C_9$

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: $C_6^2:(D_4\times D_9)$

Order 2592: $C_2^2\times C_6^2.(C_3\times S_3)$, $C_6^3.D_6$, $C_6^2:D_{36}$, $C_3^3.\GL(2,\mathbb{Z}/4)$, $C_6^3.D_6$, $C_3^3.\GL(2,\mathbb{Z}/4)$, $C_6^2:(D_4\times C_9)$

Order 1728: $C_3^3.D_4^2$, $(C_2^2\times D_6):D_{18}$

Order 1296: $C_6^2:D_{18}$ x 4, $C_6^2:D_{18}$ x 2, $C_6^2:D_{18}$, $C_6^2:C_{36}$, $C_6^3.S_3$, $C_6^3.C_6$, $C_6^3.C_6$, $C_6^2:D_{18}$

Order 864: $C_6^3:C_2^2$ x 2, $C_6^3:C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2.S_4$ x 2, $C_6^2.S_4$, $C_2^2:D_{18}\times S_3$, $C_6^3.C_2^2$, $C_6^3:C_2^2$, $C_6^3.C_2^2$, $(C_2^2\times D_6):C_{18}$, $C_3^2.\GL(2,\mathbb{Z}/4)$, $C_6^2.D_{12}$, $C_6.(S_3\times S_4)$

Order 648: $C_6^2:D_9$ x 3, $C_6^2:C_{18}$ x 2, $C_3^2:D_{36}$, $(C_3\times C_6).D_{18}$, $C_3^2:(C_4\times D_9)$, $C_6^2:D_9$, $C_2\times C_3^2:D_{18}$, $C_6^2:C_{18}$, $C_6^2:D_9$, $C_6^2:D_9$, $C_6^2:C_{18}$

Order 576: $C_3^2:D_4^2$ x 2, $C_2^4:S_3^2$, $C_2^4:D_{18}$

Order 432: $C_6^3:C_2$ x 15, $C_6^2:D_6$ x 8, $C_6^2.D_6$ x 5, $C_6^2.D_6$ x 5, $C_6^2.D_6$ x 4, $C_6^2:C_{12}$ x 3, $C_6^3:C_2$ x 2, $C_6^2:C_{12}$ x 2, $C_6^2.D_6$ 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, $C_3\times C_6.S_4$ x 2, $C_2\times C_6^3$, $C_3:D_6^2$, $C_2^2:C_9\times D_6$, $C_6^2.D_6$, $D_6:D_{18}$, $C_6^2.C_{12}$

Order 324: $C_3^2:D_{18}$ x 4, $C_3^2:D_{18}$ x 2, $C_6^2:C_9$, $C_3^2:D_{18}$, $C_3^2:C_{36}$, $C_2\times C_3^2:C_{18}$, $C_6^2:C_9$, $(C_3\times C_{18}).S_3$

Order 288: $C_6^2:D_4$ x 5, $D_6^2:C_2$ x 4, $C_6^2:D_4$ x 4, $C_6^2:D_4$ x 3, $D_6^2:C_2$ x 3, $C_2^3.S_3^2$ x 3, $C_2^4:D_9$ x 3, $D_6^2:C_2$ x 2, $C_2^3:D_{18}$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ 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, $C_6^2:D_4$, $C_2^4:C_{18}$, $C_2^2:D_{36}$, $C_{12}.S_4$

Order 216: $C_6^2:C_6$ x 20, $C_6^2:C_6$ x 12, $C_6^2.C_6$ x 9, $C_3^2:D_{12}$ x 9, $C_6^3$ x 5, $C_6:S_3^2$ x 5, $C_6.S_3^2$ x 4, $C_6^2.C_6$ x 4, $C_3^2.S_4$ x 3, $C_3^2.S_4$ x 2, $D_{18}:C_6$ x 2, $C_6^2.C_6$, $D_6:C_{18}$, $C_9:D_{12}$, $C_3:D_{36}$, $C_6.D_{18}$, $C_6:D_{18}$, $S_3\times D_{18}$

Order 192: $C_2^4:D_6$ x 2, $C_2^4:D_6$

Order 162: $C_3^2:D_9$ x 3, $C_3^2:C_{18}$ x 2, $C_3^2:C_{18}$, $C_3^2:D_9$

Order 144: $C_6^2:C_2^2$ x 49, $D_6:D_6$ x 16, $C_6^2:C_2^2$ x 15, $C_6.D_{12}$ x 10, $C_6:D_{12}$ x 10, $C_6^2:C_4$ x 9, $D_6:D_6$ x 8, $C_2^2:D_{18}$ x 8, $C_6.D_{12}$ x 5, $C_2^2\times C_6^2$ x 5, $C_6^2:C_2^2$ x 5, $C_6^2:C_4$ x 5, $D_6:D_6$ x 5, $C_6.D_{12}$ x 4, $D_6.D_6$ x 4, $C_3^2:C_4^2$ x 3, $D_6^2$ x 3, $C_2^3:C_{18}$ x 3, $C_6^2:C_4$ x 3, $C_2^2.S_3^2$ x 2, $C_6.S_4$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_4$ x 2, $C_2^2.S_3^2$ x 2, $D_4\times D_9$, $C_2^2:C_{36}$

Order 108: $C_3^2:D_6$ x 20, $C_3\times C_6^2$ x 9, $C_3^2:C_{12}$ x 8, $C_3:S_3^2$ x 4, $C_3:D_{18}$ x 4, $C_9:C_{12}$ x 2, $C_6\times C_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3^2.A_4$ x 2, $S_3\times D_9$ x 2, $C_3:C_{36}$, $S_3\times C_{18}$

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

Order 81: $C_3^2:C_9$

Order 72: $C_6\wr C_2$ x 67, $C_6\times D_6$ x 38, $C_2\times C_6^2$ x 34, $C_6:C_{12}$ x 29, $C_6^2:C_2$ x 20, $C_3:D_{12}$ x 18, $S_3\times D_6$ x 15, $C_6:D_6$ x 12, $C_6.D_6$ x 9, $D_6:S_3$ x 9, $C_6.D_6$ x 8, $C_2^2:C_{18}$ x 5, $C_2^2:D_9$ x 5, $C_6.D_6$ x 4, $C_9:D_4$ x 3, $C_2\times D_{18}$ x 2, $D_{36}$, $C_4\times D_9$, $D_4\times C_9$

Order 64: $D_4^2$

Order 54: $C_3^2:C_6$ x 6, $C_3^2\times C_6$ x 5, $C_3\times C_{18}$ x 4, $C_3:D_9$ x 3, $C_3\times D_9$ x 2, $S_3\times C_9$

Order 48: $C_6:D_4$ x 67, $S_3\times D_4$ x 16, $C_6\times D_4$ x 15, $C_6.D_4$ x 14, $D_6:C_4$ x 10, $C_2^2\times D_6$ x 8, $C_2^3\times C_6$ x 7, $C_6.C_2^3$ x 7, $C_2\times D_{12}$ x 5, $C_4\times D_6$ x 4, $C_6.D_4$ x 4, $C_2^2:C_{12}$ x 3, $C_{12}:C_4$ x 3, $C_2^2\times C_{12}$ x 2, $C_{12}:C_4$ x 2

Order 36: $C_6^2$ x 67, $C_6\times S_3$ x 64, $C_3:C_{12}$ x 24, $C_6:S_3$ x 20, $S_3^2$ x 12, $C_3^2:C_4$ x 8, $D_{18}$ x 8, $C_2\times C_{18}$ x 3, $C_2^2:C_9$ x 2, $C_9:C_4$ x 2, $C_{36}$

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

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

Order 24: $C_3:D_4$ x 85, $C_2\times D_6$ x 53, $C_2^2\times C_6$ x 46, $C_6:C_4$ x 33, $C_3\times D_4$ x 20, $C_2\times C_{12}$ x 9, $D_{12}$ x 9, $C_4\times S_3$ x 8

Order 18: $C_3\times C_6$ x 34, $C_3\times S_3$ x 18, $C_3:S_3$ x 6, $C_{18}$ x 5, $D_9$ x 5

Order 16: $C_2\times D_4$ x 28, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 87, $D_6$ x 76, $C_3:C_4$ x 24, $C_{12}$ x 8

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

Order 8: $D_4$ x 29, $C_2^3$ x 22, $C_2\times C_4$ x 13

Order 6: $C_6$ x 40, $S_3$ x 18

Order 4: $C_2^2$ x 33, $C_4$ x 8

Order 3: $C_3$ x 5

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $C_6^2:(D_4\times D_9)$

Order 2592: $C_2^2\times C_6^2.(C_3\times S_3)$, $C_6^3.D_6$, $C_6^2:D_{36}$, $C_3^3.\GL(2,\mathbb{Z}/4)$, $C_6^3.D_6$, $C_3^3.\GL(2,\mathbb{Z}/4)$, $C_6^2:(D_4\times C_9)$

Order 1728: $C_3^3.D_4^2$, $(C_2^2\times D_6):D_{18}$

Order 1296: $C_6^2:D_{18}$ x 3, $C_6^2:D_{18}$, $C_6^2:C_{36}$, $C_6^3.S_3$, $C_6^3.C_6$, $C_6^3.C_6$, $C_6^2:D_{18}$, $C_6^2:D_{18}$

Order 864: $C_6^3:C_2^2$ x 2, $C_6^3:C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2.S_4$ x 2, $C_6^2.S_4$, $C_2^2:D_{18}\times S_3$, $C_6^3.C_2^2$, $C_6^3:C_2^2$, $C_6^3.C_2^2$, $(C_2^2\times D_6):C_{18}$, $C_3^2.\GL(2,\mathbb{Z}/4)$, $C_6^2.D_{12}$, $C_6.(S_3\times S_4)$

Order 648: $C_6^2:C_{18}$ x 2, $C_6^2:D_9$ x 2, $C_3^2:D_{36}$, $(C_3\times C_6).D_{18}$, $C_3^2:(C_4\times D_9)$, $C_6^2:D_9$, $C_2\times C_3^2:D_{18}$, $C_6^2:C_{18}$, $C_6^2:D_9$, $C_6^2:D_9$, $C_6^2:C_{18}$

Order 576: $C_3^2:D_4^2$ x 2, $C_2^4:S_3^2$, $C_2^4:D_{18}$

Order 432: $C_6^3:C_2$ x 13, $C_6^2:D_6$ x 6, $C_6^2.D_6$ x 5, $C_6^2.D_6$ x 5, $C_6^2.D_6$ x 3, $C_6^2:C_{12}$ x 3, $C_6^3:C_2$ x 2, $C_6^2:C_{12}$ x 2, $C_6^2.D_6$ x 2, $C_6^2.A_4$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_3\times C_6.S_4$ x 2, $C_2\times C_6^3$, $C_3:D_6^2$, $C_2^2:C_9\times D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $D_6:D_{18}$, $C_6^2.C_{12}$

Order 324: $C_3^2:D_{18}$ x 3, $C_6^2:C_9$, $C_3^2:D_{18}$, $C_3^2:C_{36}$, $C_2\times C_3^2:C_{18}$, $C_6^2:C_9$, $C_3^2:D_{18}$, $(C_3\times C_{18}).S_3$

Order 288: $C_6^2:D_4$ x 5, $D_6^2:C_2$ x 4, $C_6^2:D_4$ x 4, $C_6^2:D_4$ x 3, $D_6^2:C_2$ x 3, $C_2^3.S_3^2$ x 3, $C_2^4:D_9$ x 3, $D_6^2:C_2$ x 2, $C_2^3:D_{18}$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ 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, $C_6^2:D_4$, $C_2^4:C_{18}$, $C_2^2:D_{36}$, $C_{12}.S_4$

Order 216: $C_6^2:C_6$ x 16, $C_6^2:C_6$ x 11, $C_6^2.C_6$ x 8, $C_3^2:D_{12}$ x 7, $C_6^3$ x 5, $C_6:S_3^2$ x 4, $C_6^2.C_6$ x 4, $C_6.S_3^2$ x 3, $C_3^2.S_4$ x 2, $C_3^2.S_4$ x 2, $D_{18}:C_6$ x 2, $C_6^2.C_6$, $D_6:C_{18}$, $C_9:D_{12}$, $C_3:D_{36}$, $C_6.D_{18}$, $C_6:D_{18}$, $S_3\times D_{18}$

Order 192: $C_2^4:D_6$ x 2, $C_2^4:D_6$

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

Order 144: $C_6^2:C_2^2$ x 39, $C_6^2:C_2^2$ x 13, $D_6:D_6$ x 12, $C_6.D_{12}$ x 10, $C_6:D_{12}$ x 10, $C_6^2:C_4$ x 9, $D_6:D_6$ x 6, $C_2^2:D_{18}$ x 6, $C_6.D_{12}$ x 5, $C_2^2\times C_6^2$ x 5, $C_6^2:C_2^2$ x 5, $C_6^2:C_4$ x 5, $D_6:D_6$ x 5, $C_6.D_{12}$ x 4, $D_6.D_6$ x 4, $C_3^2:C_4^2$ x 3, $D_6^2$ x 3, $C_2^3:C_{18}$ x 3, $C_6^2:C_4$ x 3, $C_2^2.S_3^2$ x 2, $C_6.S_4$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_4$ x 2, $C_2^2.S_3^2$ x 2, $D_4\times D_9$, $C_2^2:C_{36}$

Order 108: $C_3^2:D_6$ x 17, $C_3\times C_6^2$ x 9, $C_3^2:C_{12}$ x 7, $C_3:D_{18}$ x 3, $C_9:C_{12}$ x 2, $C_3:S_3^2$ x 2, $C_6\times C_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3^2.A_4$ x 2, $C_3:C_{36}$, $S_3\times C_{18}$, $S_3\times D_9$

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

Order 81: $C_3^2:C_9$

Order 72: $C_6\wr C_2$ x 48, $C_6\times D_6$ x 32, $C_2\times C_6^2$ x 31, $C_6:C_{12}$ x 24, $C_6^2:C_2$ x 16, $C_3:D_{12}$ x 14, $S_3\times D_6$ x 12, $C_6:D_6$ x 11, $C_6.D_6$ x 8, $D_6:S_3$ x 7, $C_6.D_6$ x 6, $C_2^2:C_{18}$ x 5, $C_2^2:D_9$ x 4, $C_9:D_4$ x 3, $C_6.D_6$ x 3, $C_2\times D_{18}$ x 2, $D_{36}$, $C_4\times D_9$, $D_4\times C_9$

Order 64: $D_4^2$

Order 54: $C_3^2\times C_6$ x 5, $C_3^2:C_6$ x 5, $C_3\times C_{18}$ x 4, $C_3:D_9$ x 2, $C_3\times D_9$ x 2, $S_3\times C_9$

Order 48: $C_6:D_4$ x 55, $C_6.D_4$ x 14, $C_6\times D_4$ x 13, $S_3\times D_4$ x 12, $D_6:C_4$ x 10, $C_2^2\times D_6$ x 8, $C_2^3\times C_6$ x 7, $C_6.C_2^3$ x 7, $C_2\times D_{12}$ x 5, $C_4\times D_6$ x 4, $C_6.D_4$ x 4, $C_2^2:C_{12}$ x 3, $C_{12}:C_4$ x 3, $C_2^2\times C_{12}$ x 2, $C_{12}:C_4$ x 2

Order 36: $C_6^2$ x 59, $C_6\times S_3$ x 50, $C_3:C_{12}$ x 20, $C_6:S_3$ x 17, $C_3^2:C_4$ x 7, $D_{18}$ x 6, $S_3^2$ x 6, $C_2\times C_{18}$ x 3, $C_2^2:C_9$ x 2, $C_9:C_4$ x 2, $C_{36}$

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

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

Order 24: $C_3:D_4$ x 62, $C_2\times D_6$ x 44, $C_2^2\times C_6$ x 42, $C_6:C_4$ x 27, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 8, $D_{12}$ x 7, $C_4\times S_3$ x 6

Order 18: $C_3\times C_6$ x 31, $C_3\times S_3$ x 14, $C_3:S_3$ x 5, $C_{18}$ x 5, $D_9$ x 4

Order 16: $C_2\times D_4$ x 24, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 76, $D_6$ x 56, $C_3:C_4$ x 20, $C_{12}$ x 7

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

Order 8: $D_4$ x 23, $C_2^3$ x 20, $C_2\times C_4$ x 11

Order 6: $C_6$ x 36, $S_3$ x 14

Order 4: $C_2^2$ x 28, $C_4$ x 7

Order 3: $C_3$ x 5

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $C_6^2:(D_4\times D_9)$ ($C_1$)

Order 2592: $C_2^2\times C_6^2.(C_3\times S_3)$ ($C_2$), $C_6^3.D_6$ ($C_2$), $C_6^2:D_{36}$ ($C_2$), $C_3^3.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6^3.D_6$ ($C_2$), $C_3^3.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6^2:(D_4\times C_9)$ ($C_2$)

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

Order 864: $C_6^3:C_2^2$ ($S_3$), $C_6^2.S_4$ ($S_3$)

Order 648: $C_6^2:D_9$ ($D_4$) x 2, $C_6^2:C_{18}$ ($C_2^3$)

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

Order 324: $C_6^2:C_9$ ($C_2\times D_4$)

Order 288: $C_6^2:D_4$ ($D_9$)

Order 216: $C_3^2.S_4$ ($C_3:D_4$) x 2, $C_6^3$ ($C_2\times D_6$), $C_6^2:C_6$ ($S_4$), $C_6^2.C_6$ ($C_2\times D_6$)

Order 144: $C_2^2\times C_6^2$ ($D_{18}$), $C_2^2\times C_6^2$ ($S_3^2$), $C_6^2:C_2^2$ ($D_{18}$), $C_6^2:C_4$ ($D_{18}$)

Order 108: $C_3\times C_6^2$ ($C_2\times S_4$), $C_3\times C_6^2$ ($S_3\times D_4$), $C_3^2:D_6$ ($C_2\times S_4$), $C_3^2:C_{12}$ ($C_2\times S_4$), $C_3^2.A_4$ ($C_6:D_4$)

Order 72: $C_2\times C_6^2$ ($S_3\times D_6$), $C_2\times C_6^2$ ($C_2\times D_{18}$), $C_6^2:C_2$ ($C_2^2:D_9$)

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

Order 48: $C_2^3\times C_6$ ($C_3^2:D_6$), $C_2^3\times C_6$ ($S_3\times D_9$)

Order 36: $C_3^2:C_4$ ($C_2^2:D_{18}$), $C_6^2$ ($D_4\times D_9$), $C_6^2$ ($S_3\times S_4$), $C_6^2$ ($D_6:D_6$), $C_6^2$ ($C_2^2:D_{18}$), $C_6:S_3$ ($C_2^2:D_{18}$)

Order 27: $C_3^3$ ($D_4\times S_4$)

Order 24: $C_2^2\times C_6$ ($C_6.S_3^2$), $C_2^2\times C_6$ ($S_3\times D_{18}$)

Order 18: $C_3\times C_6$ ($C_2^3:D_{18}$), $C_3\times C_6$ ($D_6\times S_4$)

Order 16: $C_2^4$ ($C_3^2:D_{18}$)

Order 12: $C_2\times C_6$ ($C_6^2:D_6$), $C_2\times C_6$ ($C_6^2.D_6$), $C_2\times C_6$ ($C_6^2:D_6$), $C_2\times C_6$ ($D_6:D_{18}$)

Order 9: $C_3^2$ ($C_2^4:S_3^2$), $C_3^2$ ($C_2^4:D_{18}$)

Order 8: $C_2^3$ ($C_2\times C_3^2:D_{18}$)

Order 6: $C_6$ ($C_2\times C_6^2:D_6$), $C_6$ ($C_2^2:D_{18}\times S_3$)

Order 4: $C_2^2$ ($C_6^2:D_{18}$), $C_2^2$ ($C_6^2:D_{18}$)

Order 3: $C_3$ ($C_6^2:(C_2\times S_4)$), $C_3$ ($(C_2^2\times D_6):D_{18}$)

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

Order 1: $C_1$ ($C_6^2:(D_4\times D_9)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $C_6^2:(D_4\times D_9)$ ($C_1$)

Order 2592: $C_2^2\times C_6^2.(C_3\times S_3)$ ($C_2$), $C_6^3.D_6$ ($C_2$), $C_6^2:D_{36}$ ($C_2$), $C_3^3.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6^3.D_6$ ($C_2$), $C_3^3.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_6^2:(D_4\times C_9)$ ($C_2$)

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

Order 864: $C_6^3:C_2^2$ ($S_3$), $C_6^2.S_4$ ($S_3$)

Order 648: $C_6^2:C_{18}$ ($C_2^3$), $C_6^2:D_9$ ($D_4$)

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

Order 324: $C_6^2:C_9$ ($C_2\times D_4$)

Order 288: $C_6^2:D_4$ ($D_9$)

Order 216: $C_3^2.S_4$ ($C_3:D_4$), $C_6^3$ ($C_2\times D_6$), $C_6^2:C_6$ ($S_4$), $C_6^2.C_6$ ($C_2\times D_6$)

Order 144: $C_2^2\times C_6^2$ ($D_{18}$), $C_2^2\times C_6^2$ ($S_3^2$), $C_6^2:C_2^2$ ($D_{18}$), $C_6^2:C_4$ ($D_{18}$)

Order 108: $C_3\times C_6^2$ ($C_2\times S_4$), $C_3\times C_6^2$ ($S_3\times D_4$), $C_3^2:D_6$ ($C_2\times S_4$), $C_3^2:C_{12}$ ($C_2\times S_4$), $C_3^2.A_4$ ($C_6:D_4$)

Order 72: $C_2\times C_6^2$ ($S_3\times D_6$), $C_2\times C_6^2$ ($C_2\times D_{18}$), $C_6^2:C_2$ ($C_2^2:D_9$)

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

Order 48: $C_2^3\times C_6$ ($C_3^2:D_6$), $C_2^3\times C_6$ ($S_3\times D_9$)

Order 36: $C_3^2:C_4$ ($C_2^2:D_{18}$), $C_6^2$ ($D_4\times D_9$), $C_6^2$ ($S_3\times S_4$), $C_6^2$ ($D_6:D_6$), $C_6^2$ ($C_2^2:D_{18}$), $C_6:S_3$ ($C_2^2:D_{18}$)

Order 27: $C_3^3$ ($D_4\times S_4$)

Order 24: $C_2^2\times C_6$ ($C_6.S_3^2$), $C_2^2\times C_6$ ($S_3\times D_{18}$)

Order 18: $C_3\times C_6$ ($C_2^3:D_{18}$), $C_3\times C_6$ ($D_6\times S_4$)

Order 16: $C_2^4$ ($C_3^2:D_{18}$)

Order 12: $C_2\times C_6$ ($C_6^2:D_6$), $C_2\times C_6$ ($C_6^2.D_6$), $C_2\times C_6$ ($C_6^2:D_6$), $C_2\times C_6$ ($D_6:D_{18}$)

Order 9: $C_3^2$ ($C_2^4:S_3^2$), $C_3^2$ ($C_2^4:D_{18}$)

Order 8: $C_2^3$ ($C_2\times C_3^2:D_{18}$)

Order 6: $C_6$ ($C_2\times C_6^2:D_6$), $C_6$ ($C_2^2:D_{18}\times S_3$)

Order 4: $C_2^2$ ($C_6^2:D_{18}$), $C_2^2$ ($C_6^2:D_{18}$)

Order 3: $C_3$ ($C_6^2:(C_2\times S_4)$), $C_3$ ($(C_2^2\times D_6):D_{18}$)

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

Order 1: $C_1$ ($C_6^2:(D_4\times D_9)$)

Series

Derived series

$C_6^2:(D_4\times D_9)$

$\rhd$

$C_6^2:(D_4\times D_9)$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_6^2:(D_4\times D_9)$

$\rhd$

$C_6^2:(D_4\times D_9)$

$\rhd$

$C_2^2\times C_6^2.(C_3\times S_3)$

$\rhd$

$C_2^2\times C_6^2.(C_3\times S_3)$

$\rhd$

$C_6^2:D_{18}$

$\rhd$

$C_6^2:D_{18}$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3^2.A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_6^2:(D_4\times D_9)$

$\rhd$

$C_6^2:(D_4\times D_9)$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_{18}$

$\rhd$

$C_6^2:C_9$

$\rhd$

$C_6^2:C_9$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 2 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 $111 \times 111$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $81 \times 81$ rational character table.