Abstract group 46656.ee: $C_6^4.S_3^2$

• Label: 46656.ee
• Order: \( 2^{6} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^4.S_3^2$
Order:	\(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_2\times C_6^3).C_3^4.C_2^3$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
Composition factors:	$C_2$ x 6, $C_3$ 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	555	242	1620	9894	7776	18792	7776	46656
Conjugacy classes	1	6	31	5	352	6	58	3	462
Divisions	1	6	21	5	195	4	32	2	266
Autjugacy classes	1	6	19	5	139	6	31	3	210

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	12	12	12	3	134	0	289	0	462
Irr. rational chars.	4	8	4	5	30	1	85	129	266

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

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:	${\langle a, b, c, d, e, f \mid a^{6}=b^{18}=c^{6}=d^{6}=e^{6}=f^{2}=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(1,2,4,6,7,8)(3,5)(9,10,11,12,13,15,14,17,16)(18,19,20,22,24,26)(21,23,25), (1,3)(2,5)(4,6)(7,8)(10,11,13,16,17,15)(12,14)(19,21,24,26,25,20)\rangle$
Transitive group:	36T15864	36T15865			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^4.S_3)$ $\,\rtimes\,$ $S_3$	$(C_6^4.S_3)$ $\,\rtimes\,$ $C_6$	$(C_6^4.D_6)$ $\,\rtimes\,$ $C_3$	$(C_6^4.C_6)$ $\,\rtimes\,$ $C_6$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^4$ . $S_3^2$	$C_6^4$ . $(C_6\times S_3)$	$(C_3\times C_6^4)$ . $D_6$	$C_3$ . $(C_6^4.D_6)$	all 36

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:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 531284 subgroups in 23128 conjugacy classes, 56 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_6^4.S_3^2$
Commutator:	$G' \simeq$ $C_2^4.C_3^3:C_9$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3\times C_6^2$	$G/\Phi \simeq$ $C_6^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^4$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6^4.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_6^2.S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5.C_3$

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

All subgroups of index up to 96 (order at least 486) are shown, as well as all normal subgroups of any index.

Order 46656: $C_6^4.S_3^2$

Order 23328: $C_2^4.C_3^3:C_9.C_6$, $C_6^4.(C_3\times S_3)$, $C_6^4.(C_3\times C_6)$

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

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

Order 7776: $(C_3\times C_6^3).C_6.C_2$ x 2, $C_6^4.S_3$ x 2, $C_6^4.C_6$ x 2, $C_2^4.C_3^2:C_9.C_6$, $C_2^4.C_3:D_9:C_3^2$, $C_3^3.C_6^2.C_2^3$, $(C_3\times C_6^3).C_6.C_2$, $C_3^3\times (C_2^3\times C_6):S_3$, $C_3\times C_3^3.(C_6\times D_4).C_2$, $(C_3^3\times C_6^2).C_2^3$, $C_2^4.C_3^3:D_9$, $C_6^4.S_3$, $C_6^4.C_6$

Order 5832: $C_6^2.\He_3.C_6$ x 2, $C_3^5.S_4$ x 2, $C_3^4.(C_6\times A_4)$

Order 5184: $C_6\wr C_2^2$ x 4, $C_6^3.C_6.C_2^2$ x 2, $C_3^4.C_2\wr C_2^2$, $(C_2^2\times C_6^2):D_{18}$, $C_6^2.(C_6\times S_4)$

Order 3888: $(C_3\times C_6^2).C_6^2$ x 2, $C_3^4.(C_6\times D_4)$ x 2, $C_3^3.C_6^2.C_2^2$ x 2, $C_3^2\times C_6\times C_6^2:C_2$ x 2, $C_6^4.C_3$ x 2, $C_2^4.C_3^3:C_9$ x 2, $(C_2\times C_6^3).C_3^2$ x 2, $C_3^3\times C_3:(C_6.D_4)$ x 2, $C_3\times (C_3\times C_6^2).C_6.C_2$, $C_3\times C_6^4$, $C_3^4.C_{12}.C_2^2$, $C_6\times C_3^4:D_4$, $C_2\times C_3^3.C_6^2.C_2$, $(C_3^4\times C_6).C_2^3$, $C_3^4.(C_6\times D_4)$, $(C_3\times C_6^2):D_{18}$, $C_3^3.(C_6\times S_4)$

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

Order 2592: $C_6^3.D_6$ x 12, $C_3\times (C_3\times C_6^2).C_2^3$ x 11, $C_3^2\times (C_2^3\times C_6):S_3$ x 10, $C_2^4.C_3:D_9:C_3$ x 5, $C_3^3.(C_6\times D_4).C_2$ x 4, $C_6^3:D_6$ x 4, $C_3^3\times C_2^4:S_3$ x 3, $C_6^2.C_6^2.C_2$ x 2, $C_6^3.C_6.C_2$ x 2, $(C_3^3\times C_6).C_2^4$ x 2, $C_6^3.C_6.C_2$ x 2, $(C_3\times C_6^3).C_2^2$ x 2, $C_2^4:C_9.C_3^2.C_2$ x 2, $C_6^3.C_6.C_2$ x 2, $(C_2^2\times C_6^2):C_{18}$ x 2, $C_3\times C_3^2.C_2^4:S_3$, $C_3^4.D_4.C_2^2$, $(C_3\times C_6^3).C_2^2$, $(C_2^2\times C_6^2):D_9$, $C_6^2.(C_6\times A_4)$, $C_3\times C_6^2.S_4$

Order 1944: $C_3^5:D_4$ x 6, $C_3^4.S_4$ x 5, $C_3^4.S_4$ x 5, $C_3^5:D_4$ x 4, $C_3^2\times C_6^3$ x 3, $C_6\times C_3^3:C_{12}$ x 3, $(C_3\times C_6^2):C_{18}$ x 2, $C_3^4.S_4$ x 2, $C_3^5:C_2^3$ x 2, $C_6\times C_3^3:C_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $C_3^4.S_4$ x 2, $C_3^2:C_6^3$, $C_3^5:C_2^3$, $C_3^5:Q_8$, $C_3^4:D_{12}$, $C_3^3:(S_3\times C_{12})$, $C_3^3.(C_6\times A_4)$

Order 1728: $C_6^3.C_2^3$ x 9, $C_6^3.C_2^3$ x 4, $C_6^3.C_2^3$ x 2, $C_6^3:D_4$, $C_6^2.(C_2\times S_4)$, $C_2^4:(S_3\times D_9)$

Order 1458: $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.C_6$

Order 1296: $C_6^3:C_6$ x 53, $C_6^3.C_6$ x 53, $C_6^4$ x 21, $C_6^2:S_3^2$ x 12, $C_6^2.S_3^2$ x 12, $C_6^3:C_6$ x 10, $C_6^3.C_6$ x 10, $C_6^3:C_6$ x 7, $C_6^3.C_6$ x 7, $(C_2^2\times C_6^2):C_9$ x 7, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_3\times C_6^2.A_4$ x 2, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2:D_{18}$

Order 972: $C_3^4.A_4$ x 15, $C_3^3\times C_6^2$ x 8, $C_3^4.A_4$ x 7, $C_3^3:C_6^2$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:D_{18}$

Order 864: $C_6^3.C_2^2$ x 54, $C_6^3.C_2^2$ x 45, $C_6^2.D_{12}$ x 18, $C_6^2.D_{12}$ x 12, $C_6^3:C_2^2$ x 9, $C_6^2.D_{12}$ x 9, $C_6^3:C_2^2$ x 4, $C_6^2.S_4$ x 4, $C_2^4:(S_3\times C_9)$ x 2, $C_6^3:C_2^2$ x 2, $C_6^2.S_4$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^3:C_2^2$, $C_6^2.S_4$, $C_6^3:C_2^2$, $C_6^3:C_4$, $(C_2\times C_6^2).A_4$

Order 729: $C_3^5.C_3$

Order 648: $C_3\times C_6^3$ x 181, $C_3^4:D_4$ x 106, $C_6\times C_3^2:C_{12}$ x 53, $C_3^4:C_2^3$ x 23, $C_3^4:D_4$ x 20, $C_3^4:D_4$ x 14, $C_6^2:D_9$ x 14, $C_3^3:D_{12}$ x 12, $C_3\times C_6.S_3^2$ x 12, $C_2\times C_3^3:C_{12}$ x 10, $C_6\times C_3^2:C_{12}$ x 7, $C_3^4:C_2^3$ x 5, $C_3^3.S_4$ x 5, $C_3^4:C_2^3$ x 4, $C_3^4:Q_8$ x 4, $C_3:C_6^3$ x 3, $C_3^4:C_2^3$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3\times C_6.S_3^2$ x 2, $C_3\times C_6.S_3^2$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3^3.S_4$ x 2, $C_6^2:C_{18}$ x 2, $C_6^2:D_9$ x 2, $C_3^4:C_2^3$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $S_3\times C_3^2.A_4$

Order 576: $C_2^4:S_3^2$ x 9, $C_3\times C_2^4:D_6$ x 6, $(C_6\times C_{12}):D_4$, $C_2^4:D_{18}$

Order 486: $C_3^4\times C_6$ x 3, $C_3^4:C_6$ x 2, $C_3^3:D_9$ x 2, $C_3^3:C_{18}$ x 2, $C_3^4.S_3$ x 2, $C_3^3:D_9$, $C_3^4:C_6$, $C_3^4.S_3$, $C_3^4.C_6$

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

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

Order 243: $C_3^5$

Order 216: $C_3^2.S_4$ x 6, $C_6^2:S_3$

Order 144: $C_2^2\times C_6^2$ x 3

Order 108: $C_3\times C_6^2$ x 3, $C_{18}:C_6$

Order 81: $C_3^4$ x 2

Order 64: $C_2\wr C_2^2$

Order 54: $C_9:C_6$ x 4

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

Order 36: $C_6^2$ x 3, $C_6\times S_3$ x 3

Order 27: $C_3^3$ x 3

Order 18: $C_3\times S_3$ x 21

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 3, $D_6$ x 3

Order 9: $C_3^2$ x 3

Order 6: $S_3$ x 9, $C_6$ x 8

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 12

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 96 (order at least 486) are shown, as well as all normal subgroups of any index.

Order 46656: $C_6^4.S_3^2$

Order 23328: $C_2^4.C_3^3:C_9.C_6$, $C_6^4.(C_3\times S_3)$, $C_6^4.(C_3\times C_6)$

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

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

Order 7776: $(C_3\times C_6^3).C_6.C_2$ x 2, $C_6^4.S_3$ x 2, $C_6^4.C_6$ x 2, $C_2^4.C_3^2:C_9.C_6$, $C_2^4.C_3:D_9:C_3^2$, $C_3^3.C_6^2.C_2^3$, $(C_3\times C_6^3).C_6.C_2$, $C_3^3\times (C_2^3\times C_6):S_3$, $C_3\times C_3^3.(C_6\times D_4).C_2$, $(C_3^3\times C_6^2).C_2^3$, $C_2^4.C_3^3:D_9$, $C_6^4.S_3$, $C_6^4.C_6$

Order 5832: $C_6^2.\He_3.C_6$ x 2, $C_3^5.S_4$ x 2, $C_3^4.(C_6\times A_4)$

Order 5184: $C_6^3.C_6.C_2^2$ x 2, $C_6\wr C_2^2$ x 2, $C_3^4.C_2\wr C_2^2$, $(C_2^2\times C_6^2):D_{18}$, $C_6^2.(C_6\times S_4)$

Order 3888: $(C_3\times C_6^2).C_6^2$ x 2, $C_3^4.(C_6\times D_4)$ x 2, $C_3^3.C_6^2.C_2^2$ x 2, $C_3^2\times C_6\times C_6^2:C_2$ x 2, $C_6^4.C_3$ x 2, $C_2^4.C_3^3:C_9$ x 2, $(C_2\times C_6^3).C_3^2$ x 2, $C_3^3\times C_3:(C_6.D_4)$ x 2, $C_3\times (C_3\times C_6^2).C_6.C_2$, $C_3\times C_6^4$, $C_3^4.C_{12}.C_2^2$, $C_6\times C_3^4:D_4$, $C_2\times C_3^3.C_6^2.C_2$, $(C_3^4\times C_6).C_2^3$, $C_3^4.(C_6\times D_4)$, $(C_3\times C_6^2):D_{18}$, $C_3^3.(C_6\times S_4)$

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

Order 2592: $C_3^2\times (C_2^3\times C_6):S_3$ x 6, $C_6^3.D_6$ x 6, $C_3\times (C_3\times C_6^2).C_2^3$ x 5, $C_2^4.C_3:D_9:C_3$ x 5, $C_3^3.(C_6\times D_4).C_2$ x 4, $C_3^3\times C_2^4:S_3$ x 3, $C_6^2.C_6^2.C_2$ x 2, $C_6^3.C_6.C_2$ x 2, $(C_3^3\times C_6).C_2^4$ x 2, $C_6^3.C_6.C_2$ x 2, $(C_3\times C_6^3).C_2^2$ x 2, $C_2^4:C_9.C_3^2.C_2$ x 2, $C_6^3.C_6.C_2$ x 2, $(C_2^2\times C_6^2):C_{18}$ x 2, $C_6^3:D_6$ x 2, $C_3\times C_3^2.C_2^4:S_3$, $C_3^4.D_4.C_2^2$, $(C_3\times C_6^3).C_2^2$, $(C_2^2\times C_6^2):D_9$, $C_6^2.(C_6\times A_4)$, $C_3\times C_6^2.S_4$

Order 1944: $C_3^5:D_4$ x 6, $C_3^4.S_4$ x 4, $C_3^5:D_4$ x 4, $C_3^4.S_4$ x 4, $C_3^2\times C_6^3$ x 3, $C_6\times C_3^3:C_{12}$ x 3, $(C_3\times C_6^2):C_{18}$ x 2, $C_3^4.S_4$ x 2, $C_3^5:C_2^3$ x 2, $C_6\times C_3^3:C_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $C_3^4.S_4$ x 2, $C_3^2:C_6^3$, $C_3^5:C_2^3$, $C_3^5:Q_8$, $C_3^4:D_{12}$, $C_3^3:(S_3\times C_{12})$, $C_3^3.(C_6\times A_4)$

Order 1728: $C_6^3.C_2^3$ x 5, $C_6^3.C_2^3$ x 2, $C_6^3.C_2^3$ x 2, $C_6^3:D_4$, $C_6^2.(C_2\times S_4)$, $C_2^4:(S_3\times D_9)$

Order 1458: $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.C_6$

Order 1296: $C_6^3:C_6$ x 25, $C_6^3.C_6$ x 25, $C_6^4$ x 13, $C_6^3:C_6$ x 9, $C_6^3.C_6$ x 9, $(C_2^2\times C_6^2):C_9$ x 7, $C_6^3:C_6$ x 6, $C_6^2:S_3^2$ x 6, $C_6^2.S_3^2$ x 6, $C_6^3.C_6$ x 6, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_3\times C_6^2.A_4$ x 2, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2:D_{18}$

Order 972: $C_3^4.A_4$ x 12, $C_3^3\times C_6^2$ x 8, $C_3^4.A_4$ x 6, $C_3^3:C_6^2$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:D_{18}$

Order 864: $C_6^3.C_2^2$ x 30, $C_6^3.C_2^2$ x 25, $C_6^2.D_{12}$ x 10, $C_6^2.D_{12}$ x 6, $C_6^3:C_2^2$ x 5, $C_6^2.D_{12}$ x 5, $C_6^2.S_4$ x 4, $C_2^4:(S_3\times C_9)$ x 2, $C_6^3:C_2^2$ x 2, $C_6^3:C_2^2$ x 2, $C_6^2.S_4$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^3:C_2^2$, $C_6^2.S_4$, $C_6^3:C_2^2$, $C_6^3:C_4$, $(C_2\times C_6^2).A_4$

Order 729: $C_3^5.C_3$

Order 648: $C_3\times C_6^3$ x 75, $C_3^4:D_4$ x 50, $C_6\times C_3^2:C_{12}$ x 25, $C_3^4:D_4$ x 18, $C_3^4:C_2^3$ x 13, $C_3^4:D_4$ x 12, $C_6^2:D_9$ x 10, $C_2\times C_3^3:C_{12}$ x 9, $C_6\times C_3^2:C_{12}$ x 6, $C_3^3:D_{12}$ x 6, $C_3\times C_6.S_3^2$ x 6, $C_3^4:C_2^3$ x 5, $C_3^3.S_4$ x 4, $C_3:C_6^3$ x 3, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^4:Q_8$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3\times C_6.S_3^2$ x 2, $C_3\times C_6.S_3^2$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3^3.S_4$ x 2, $C_6^2:C_{18}$ x 2, $C_6^2:D_9$ x 2, $C_3^4:C_2^3$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $S_3\times C_3^2.A_4$

Order 576: $C_2^4:S_3^2$ x 5, $C_3\times C_2^4:D_6$ x 4, $(C_6\times C_{12}):D_4$, $C_2^4:D_{18}$

Order 486: $C_3^4\times C_6$ x 3, $C_3^4:C_6$ x 2, $C_3^3:D_9$ x 2, $C_3^3:C_{18}$ x 2, $C_3^4.S_3$ x 2, $C_3^3:D_9$, $C_3^4:C_6$, $C_3^4.S_3$, $C_3^4.C_6$

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

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

Order 243: $C_3^5$

Order 216: $C_3^2.S_4$ x 4, $C_6^2:S_3$

Order 144: $C_2^2\times C_6^2$ x 3

Order 108: $C_3\times C_6^2$ x 3, $C_{18}:C_6$

Order 81: $C_3^4$ x 2

Order 64: $C_2\wr C_2^2$

Order 54: $C_9:C_6$ x 4

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

Order 36: $C_6^2$ x 3, $C_6\times S_3$

Order 27: $C_3^3$ x 3

Order 18: $C_3\times S_3$ x 7

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 3, $D_6$

Order 9: $C_3^2$ x 3

Order 6: $C_6$ x 6, $S_3$ x 3

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 8

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 46656: $C_6^4.S_3^2$ ($C_1$)

Order 23328: $C_2^4.C_3^3:C_9.C_6$ ($C_2$), $C_6^4.(C_3\times S_3)$ ($C_2$), $C_6^4.(C_3\times C_6)$ ($C_2$)

Order 15552: $C_6^4.D_6$ ($C_3$)

Order 11664: $C_6^4.C_3^2$ ($C_2^2$)

Order 7776: $C_3\times C_3^3.(C_6\times D_4).C_2$ ($S_3$), $C_2^4.C_3^3:D_9$ ($C_6$), $C_6^4.S_3$ ($C_6$), $C_6^4.S_3$ ($S_3$), $C_6^4.C_6$ ($C_6$)

Order 3888: $C_3\times C_6^4$ ($D_6$), $C_2^4.C_3^3:C_9$ ($C_2\times C_6$), $(C_2\times C_6^3).C_3^2$ ($D_6$)

Order 2592: $C_3\times C_3^2.C_2^4:S_3$ ($C_3\times S_3$), $(C_2^2\times C_6^2):D_9$ ($C_3\times S_3$)

Order 1944: $C_3^5:C_2^3$ ($S_4$)

Order 1296: $C_6^4$ ($C_6\times S_3$), $C_6^4$ ($S_3^2$), $(C_2^2\times C_6^2):C_9$ ($C_6\times S_3$)

Order 972: $C_3^3\times C_6^2$ ($C_2\times S_4$)

Order 864: $C_6^3.C_2^2$ ($C_9:C_6$)

Order 648: $C_3^4:C_2^3$ ($C_3\times S_4$)

Order 432: $C_2\times C_6^3$ ($C_3\times S_3^2$), $C_2\times C_6^3$ ($C_{18}:C_6$), $C_2\times C_6^3$ ($C_3^2:D_6$)

Order 324: $C_3^2\times C_6^2$ ($C_6\times S_4$), $C_3^2\times C_6^2$ ($S_3\times S_4$)

Order 243: $C_3^5$ ($C_2^3:S_4$)

Order 216: $C_6^2:S_3$ ($C_3^2.S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_3^3:D_6$), $C_2^2\times C_6^2$ ($C_3^2.S_3^2$), $C_2^2\times C_6^2$ ($C_3^3:D_6$)

Order 108: $C_3\times C_6^2$ ($C_6^2:D_6$), $C_3\times C_6^2$ ($C_6^2.D_6$), $C_3\times C_6^2$ ($C_6^2:D_6$)

Order 81: $C_3^4$ ($C_3\times C_2^3:S_4$), $C_3^4$ ($C_2^4:S_3^2$)

Order 48: $C_2^3\times C_6$ ($C_3^4:D_6$), $C_2^3\times C_6$ ($C_3^3.S_3^2$)

Order 36: $C_6^2$ ($C_6^2.S_3^2$), $C_6^2$ ($C_6^2.S_3^2$), $C_6^2$ ($C_6^2.S_3^2$)

Order 27: $C_3^3$ ($C_3\times C_2^4:S_3^2$), $C_3^3$ ($C_6^2.(C_2\times S_4)$), $C_3^3$ ($(C_2^2\times C_6^2):D_6$)

Order 16: $C_2^4$ ($C_3^5.D_6$)

Order 12: $C_2\times C_6$ ($C_3^3:(C_6\times S_4)$), $C_2\times C_6$ ($C_3^3.(C_6\times S_4)$)

Order 9: $C_3^2$ ($(C_2\times C_6^3):D_6$), $C_3^2$ ($(C_2\times C_6^3):D_6$), $C_3^2$ ($C_6^2.(C_6\times S_4)$)

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

Order 3: $C_3$ ($C_6^4:D_6$), $C_3$ ($C_6^4.D_6$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 46656: $C_6^4.S_3^2$ ($C_1$)

Order 23328: $C_2^4.C_3^3:C_9.C_6$ ($C_2$), $C_6^4.(C_3\times S_3)$ ($C_2$), $C_6^4.(C_3\times C_6)$ ($C_2$)

Order 15552: $C_6^4.D_6$ ($C_3$)

Order 11664: $C_6^4.C_3^2$ ($C_2^2$)

Order 7776: $C_3\times C_3^3.(C_6\times D_4).C_2$ ($S_3$), $C_2^4.C_3^3:D_9$ ($C_6$), $C_6^4.S_3$ ($C_6$), $C_6^4.S_3$ ($S_3$), $C_6^4.C_6$ ($C_6$)

Order 3888: $C_3\times C_6^4$ ($D_6$), $C_2^4.C_3^3:C_9$ ($C_2\times C_6$), $(C_2\times C_6^3).C_3^2$ ($D_6$)

Order 2592: $C_3\times C_3^2.C_2^4:S_3$ ($C_3\times S_3$), $(C_2^2\times C_6^2):D_9$ ($C_3\times S_3$)

Order 1944: $C_3^5:C_2^3$ ($S_4$)

Order 1296: $C_6^4$ ($C_6\times S_3$), $C_6^4$ ($S_3^2$), $(C_2^2\times C_6^2):C_9$ ($C_6\times S_3$)

Order 972: $C_3^3\times C_6^2$ ($C_2\times S_4$)

Order 864: $C_6^3.C_2^2$ ($C_9:C_6$)

Order 648: $C_3^4:C_2^3$ ($C_3\times S_4$)

Order 432: $C_2\times C_6^3$ ($C_3\times S_3^2$), $C_2\times C_6^3$ ($C_{18}:C_6$), $C_2\times C_6^3$ ($C_3^2:D_6$)

Order 324: $C_3^2\times C_6^2$ ($C_6\times S_4$), $C_3^2\times C_6^2$ ($S_3\times S_4$)

Order 243: $C_3^5$ ($C_2^3:S_4$)

Order 216: $C_6^2:S_3$ ($C_3^2.S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_3^3:D_6$), $C_2^2\times C_6^2$ ($C_3^2.S_3^2$), $C_2^2\times C_6^2$ ($C_3^3:D_6$)

Order 108: $C_3\times C_6^2$ ($C_6^2:D_6$), $C_3\times C_6^2$ ($C_6^2.D_6$), $C_3\times C_6^2$ ($C_6^2:D_6$)

Order 81: $C_3^4$ ($C_3\times C_2^3:S_4$), $C_3^4$ ($C_2^4:S_3^2$)

Order 48: $C_2^3\times C_6$ ($C_3^4:D_6$), $C_2^3\times C_6$ ($C_3^3.S_3^2$)

Order 36: $C_6^2$ ($C_6^2.S_3^2$), $C_6^2$ ($C_6^2.S_3^2$), $C_6^2$ ($C_6^2.S_3^2$)

Order 27: $C_3^3$ ($C_3\times C_2^4:S_3^2$), $C_3^3$ ($C_6^2.(C_2\times S_4)$), $C_3^3$ ($(C_2^2\times C_6^2):D_6$)

Order 16: $C_2^4$ ($C_3^5.D_6$)

Order 12: $C_2\times C_6$ ($C_3^3:(C_6\times S_4)$), $C_2\times C_6$ ($C_3^3.(C_6\times S_4)$)

Order 9: $C_3^2$ ($(C_2\times C_6^3):D_6$), $C_3^2$ ($(C_2\times C_6^3):D_6$), $C_3^2$ ($C_6^2.(C_6\times S_4)$)

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

Order 3: $C_3$ ($C_6^4:D_6$), $C_3$ ($C_6^4.D_6$)

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

Series

Derived series

$C_6^4.S_3^2$

$\rhd$

$C_2^4.C_3^3:C_9$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

Chief series

$C_6^4.S_3^2$

$\rhd$

$C_6^4.(C_3\times S_3)$

$\rhd$

$C_6^4.C_3^2$

$\rhd$

$C_2^4.C_3^3:C_9$

$\rhd$

$(C_2^2\times C_6^2):C_9$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6^4.S_3^2$

$\rhd$

$C_2^4.C_3^3:C_9$

Upper central series

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $462 \times 462$ 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 $266 \times 266$ rational character table (warning: may be slow to load).