Abstract group 2239488.bd: $C_6^4.D_6^2:D_6$

• Label: 2239488.bd
• Order: \( 2^{10} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_6^4.D_6^2:D_6$
Order:	\(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.C_2^4:C_3.C_2^3.C_6.C_2^6$, of order \(107495424\)\(\medspace = 2^{14} \cdot 3^{8} \)
Composition factors:	$C_2$ x 10, $C_3$ x 7
Derived length:	$3$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	58815	3320	406080	508104	20736	931392	248832	62208	2239488
Conjugacy classes	1	57	31	44	1486	2	354	10	1	1986
Divisions	1	57	31	44	1486	2	354	10	1	1986
Autjugacy classes	1	30	15	23	333	2	82	6	1	493

Minimal presentations

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

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid c^{4}=d^{6}=e^{6}=f^{6}=g^{6}=h^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,4,6,5,8,2)(3,9,14,10,13,17)(7,12)(16,18)(19,20,24,22)(21,26)(23,27)(25,28) \!\cdots\! \rangle$
Transitive group:	36T43671				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_6^6$ . $(S_3\times D_4)$ (6)	$(C_6^4.S_3^3)$ . $D_4$ (12)	$C_6^4$ . $(D_6^2:D_6)$	$C_6^4$ . $(D_6^2:D_6)$	all 96

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 207 normal subgroups (103 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_3\times C_6^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 18 (order at least 124416) are shown, as well as all normal subgroups of any index.

Order 2239488: $C_6^4.D_6^2:D_6$

Order 1119744: $C_6^4.S_3^3:C_4$ x 2, $C_3^4.(D_6^2.\GL(2,\mathbb{Z}/4))$ x 2, $C_6^4.S_3^2:D_{12}$ x 2, $C_6^4.S_3^3:C_2^2$ x 2, $C_6^4.C_3^3.C_2^5$, $C_6^4.S_3^2:S_4$, $C_6^4.S_3^2:S_4$, $C_6^2.C_6^4:D_{12}$, $C_3^6.(C_2^7.D_6)$, $C_6\wr D_6.C_2$, $C_6^4.D_6^2:C_6$

Order 746496: $C_3^6.C_2^6.C_2^4$

Order 559872: $C_6^4.C_3^3.C_2^4$ x 14, $C_3^6.C_2\wr D_6$ x 8, $C_6^4.C_3.D_6^2$ x 3, $C_2^3.(C_3^6.\GL(2,\mathbb{Z}/4))$ x 2, $C_6^4.S_3^2:D_6$ x 2, $C_6^4.S_3^2:D_6$ x 2, $C_2^3.(C_3^6.\GL(2,\mathbb{Z}/4))$ x 2, $C_3^6.C_2^5:D_{12}$ x 2, $C_3^6.(C_2^3.\GL(2,\mathbb{Z}/4))$ x 2, $C_3^6.(C_2^3.\GL(2,\mathbb{Z}/4))$ x 2, $C_3^6.(C_2^6.D_6)$ x 2, $C_3^6.C_2^7:C_6$ x 2, $C_6^4.S_3^2:C_{12}$ x 2, $C_6^4.\He_3.C_2^4$, $C_3^6.C_2^4:C_3.C_2^4$, $C_6^4.S_3^3:C_2$, $C_3^6.C_2\wr D_6$, $C_3^6.C_2\wr D_6$, $C_6^3.C_6^3:D_6$, $C_2\times C_6^4.S_3^3$, $C_6^4.D_6:S_3^2$, $C_6^4.(C_{12}.S_3^2)$, $C_3^6.C_2\wr D_6$, $C_6^2.C_6^4:D_6$, $C_3^6.(C_2^3.\GL(2,\mathbb{Z}/4))$, $C_6^4.(A_4\times S_3^2)$, $C_6^2.C_6^4:C_{12}$, $C_6\wr D_6$

Order 373248: $C_3^6.C_2^6.C_2^3$ x 30, $C_6^4.C_6^2.C_2^3$, $C_2\times C_3^6.C_2^6.C_2^2$, $C_6^4.(D_6\times S_4)$

Order 279936: $C_6^4.S_3^3$ x 36, $C_6^4.C_3^3.C_2^3$ x 27, $C_6^4.\He_3.C_2^3$ x 8, $C_3^6.C_2^4:C_3.C_2^3$ x 8, $C_3^6.C_2^2\wr S_3$ x 8, $C_3^6.C_2\wr S_3$ x 4, $C_6^2.C_3^4:\GL(2,\mathbb{Z}/4)$ x 4, $C_3^2.C_6^4:D_{12}$ x 4, $C_3^6.(C_2^5.D_6)$ x 4, $C_6^4.S_3^3$ x 4, $C_3^6.C_2\wr C_6$ x 4, $C_3^4.A_4.D_6^2.C_2$ x 2, $C_3^6.\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_3^4.D_6^2:D_{12}$ x 2, $C_3^4.S_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^6.(C_2^5.D_6)$ x 2, $C_6^2.C_3^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^6.(C_2^4.S_4)$ x 2, $C_3^6.C_2^5:C_{12}$ x 2, $(C_3\times C_6^4).C_6^2.C_2$, $C_6^2.C_3^4:\GL(2,\mathbb{Z}/4)$, $C_3^2.C_6^4:D_{12}$, $C_3^6.(C_2^5.D_6)$, $C_6^4.S_3^3$, $C_6^4.S_3^2:C_6$, $C_6^3.C_6^3:S_3$, $C_6^3.C_6^3:S_3$, $C_6^3.C_6^3:S_3$, $C_6^3.C_6^3:S_3$, $C_3^6.C_2\wr S_3$, $C_6^3.C_6^2:S_3^2$, $C_3^6.C_2\wr C_6$, $C_6^4.(C_6\times S_3^2)$, $C_6^3.C_6^3:C_6$, $C_6^3.C_6\wr S_3$, $C_6^3.C_6\wr S_3$, $C_6^4.(C_6\times S_3^2)$, $C_6^3.C_6\wr S_3$, $C_6^3.C_6\wr S_3$, $C_6^5.C_6^2$

Order 248832: $C_6^4.(D_4\times S_4)$

Order 186624: $C_3^6.C_2^6.C_2^2$ x 253, $C_2\times C_3^6.C_2^6.C_2$ x 33, $C_6^4.D_6^2$ x 28, $C_6^6.C_2^2$ x 2, $C_6^4.C_6^2.C_2^2$ x 2, $C_3.C_2^4:\He_3.D_6^2$, $C_2^3\times C_3^6.C_2^5$, $C_2\times (C_3^4\times C_6^2).C_2^5$, $(C_3^3\times C_6^3).C_2^5$, $(C_3\times C_6^4).C_6.C_2^3$, $(C_2\times C_6^3).C_3^3.C_2^4$, $C_6^4.(C_6\times S_4)$, $C_6^4.(A_4\times D_6)$, $C_6^4.(C_6\times S_4)$, $C_6^4.(C_6\times S_4)$, $C_6^3.(S_4\times S_3^2)$, $C_6^3.(S_4\times S_3^2)$, $C_6^3.(S_4\times S_3^2)$, $C_6^3.(S_4\times S_3^2)$, $C_6^3.(S_4\times S_3^2)$, $C_6^4.D_6^2$, $C_6^3.(S_4\times S_3^2)$, $C_6^3.(S_4\times S_3^2)$, $C_6^4.(S_3\times S_4)$, $C_6^4.(S_3\times S_4)$, $C_6^4.(S_3\times S_4)$

Order 139968: $(C_3\times C_6^3).S_3^3$ x 39, $C_6^4.C_3^3.C_2^2$ x 29, $(C_3^2\times C_6^4).C_6.C_2$ x 14, $C_6^4.C_3:S_3^2$ x 12, $C_6^4.C_3^2.D_6$ x 10, $C_6^4.\He_3.C_2^2$ x 8, $C_3^6.(D_4\times S_4)$ x 8, $C_2^5.C_3\wr S_3$ x 7, $(C_3\times C_6^4).C_6^2$ x 6, $C_2\times C_3^6.\GL(2,\mathbb{Z}/4)$ x 6, $C_6^4.C_3.C_6^2$ x 4, $(C_3^3\times C_6^2).D_6^2$ x 4, $(C_3^5\times C_6).\GL(2,\mathbb{Z}/4)$ x 4, $C_3^6.A_4.C_2^4$ x 3, $(C_3^4\times C_6^2).C_6.C_2^3$ x 2, $(C_3^5\times C_6).\GL(2,\mathbb{Z}/4)$ x 2, $C_3^6.C_2^3:D_{12}$ x 2, $C_3^5:S_3.\GL(2,\mathbb{Z}/4)$ x 2, $C_3^5:S_3.\GL(2,\mathbb{Z}/4)$ x 2, $C_3^6.(C_2^4.D_6)$ x 2, $C_3^6.C_2^5:C_6$ x 2, $C_3^4.D_6^2:C_{12}$ x 2, $C_6^4.C_3:S_3^2$ x 2, $C_3^5.(S_3\times \GL(2,\mathbb{Z}/4))$ x 2, $C_6^4.C_3^3:C_4$ x 2, $C_3^5:S_3.\GL(2,\mathbb{Z}/4)$ x 2, $C_6^4.(C_3\times S_3^2)$ x 2, $C_3^2.C_6^4:C_{12}$ x 2, $C_6^4.(C_3\times S_3^2)$ x 2, $C_3^4.A_4.C_3^2.C_2^4$, $(C_3^3\times C_6^2).C_6^2.C_2^2$, $(C_3^2\times C_6^2).\He_3.C_2^4$, $(C_3^2\times C_6^2).C_3^3.C_2^4$, $C_3^4.D_6^2:D_6$, $(C_3\times C_6^3).S_3^3$, $C_3^5.(S_4\times D_{12})$, $(C_3\times C_6^3).S_3^3$, $(C_3\times C_6^3).S_3^3$, $(C_3\times C_6^3).S_3^3$, $C_3^6.(D_4\times S_4)$, $C_6^4.(C_3\times S_3^2)$, $C_6^4.C_3^2:C_{12}$, $C_6^4.(C_3\times S_3^2)$, $C_2^5.C_3\wr S_3$, $C_6^4.(C_3^2\times D_6)$, $C_6^3.C_6\wr C_3$

Order 124416: $C_6^5.C_2^4$ x 5, $C_2\times C_3^5.C_2^6.C_2^2$ x 5, $C_6^4.(D_6.D_4)$ x 2, $C_6^4.(D_4\times D_6)$ x 2, $C_6^2.D_6^2:D_{12}$ x 2, $C_6^2.S_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:\He_3.D_6^2.C_2$, $(C_2\times C_6^3).C_6^2.C_2^3$, $C_6^4.\GL(2,\mathbb{Z}/4)$, $C_6^4.\GL(2,\mathbb{Z}/4)$, $C_6^4.C_4:S_4$, $C_6^4.(C_4\times S_4)$, $C_6^4.(C_2^2\times S_4)$, $C_6^4.(C_2^2\times S_4)$, $C_6^4.(D_4\times A_4)$

Order 93312: $C_3^6.C_2^6.C_2$ x 5, $C_2^2\times C_3^6.C_2^5$ x 2, $C_2^3\times C_3^6.C_2^4$

Order 69984: $C_6^4.C_3^3.C_2$ x 5, $(C_3^2\times C_6^4).C_6$ x 2

Order 46656: $C_3^6.C_2^5.C_2$ x 6, $C_2\times C_3^6.C_2^5$ x 4, $C_2^2\times C_3^6.C_2^4$ x 2, $C_6^6$, $C_3^6.C_2^6$

Order 34992: $C_6^4.C_3^3$

Order 31104: $C_6^4.C_6.C_2^2$

Order 23328: $C_3^6.C_2^5$ x 4, $C_3^6.C_2^4.C_2$ x 3, $C_2\times C_3^6.C_2^4$ x 2, $C_3\times C_6^5$

Order 15552: $C_6^4.D_6$ x 2, $C_6^4.C_6.C_2$

Order 11664: $C_3^6.C_2^4$ x 7, $C_3^2\times C_6^4$ x 2

Order 7776: $C_6^4.S_3$ x 2, $C_6^4.C_6$

Order 5832: $C_3^6.C_2^3$ x 5, $C_3^3\times C_6^3$

Order 5184: $C_2^2\times C_6^4$

Order 3888: $C_6^4.C_3$

Order 2916: $C_3^5.D_6$ x 6, $C_3^4\times C_6^2$ x 2

Order 2592: $C_2\times C_6^4$

Order 1458: $C_3^5:S_3$ x 2, $C_3^5\times C_6$

Order 1296: $C_6^4$ x 2

Order 729: $C_3^6$

Order 648: $C_3\times C_6^3$

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

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

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

Order 162: $C_3^3\times C_6$

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

Order 81: $C_3^4$

Order 72: $C_2\times C_6^2$

Order 64: $C_2^6$

Order 36: $C_6^2$ x 2

Order 32: $C_2^5$

Order 18: $C_3\times C_6$

Order 16: $C_2^4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 2239488: $C_6^4.D_6^2:D_6$ ($C_1$)

Order 1119744: $C_6^4.S_3^3:C_4$ ($C_2$) x 2, $C_3^4.(D_6^2.\GL(2,\mathbb{Z}/4))$ ($C_2$) x 2, $C_6^4.S_3^2:D_{12}$ ($C_2$) x 2, $C_6^4.S_3^3:C_2^2$ ($C_2$) x 2, $C_6^4.C_3^3.C_2^5$ ($C_2$), $C_6^4.S_3^2:S_4$ ($C_2$), $C_6^4.S_3^2:S_4$ ($C_2$), $C_6^2.C_6^4:D_{12}$ ($C_2$), $C_3^6.(C_2^7.D_6)$ ($C_2$), $C_6\wr D_6.C_2$ ($C_2$), $C_6^4.D_6^2:C_6$ ($C_2$)

Order 559872: $C_6^4.C_3^3.C_2^4$ ($C_2^2$) x 6, $C_2^3.(C_3^6.\GL(2,\mathbb{Z}/4))$ ($C_2^2$) x 2, $C_6^4.S_3^2:D_6$ ($C_2^2$) x 2, $C_6^4.S_3^2:D_6$ ($C_2^2$) x 2, $C_2^3.(C_3^6.\GL(2,\mathbb{Z}/4))$ ($C_2^2$) x 2, $C_3^6.C_2^5:D_{12}$ ($C_2^2$) x 2, $C_3^6.(C_2^3.\GL(2,\mathbb{Z}/4))$ ($C_2^2$) x 2, $C_3^6.(C_2^3.\GL(2,\mathbb{Z}/4))$ ($C_2^2$) x 2, $C_3^6.(C_2^6.D_6)$ ($C_2^2$) x 2, $C_3^6.C_2^7:C_6$ ($C_2^2$) x 2, $C_6^4.S_3^2:C_{12}$ ($C_2^2$) x 2, $C_3^6.C_2^4:C_3.C_2^4$ ($C_2^2$), $C_6^3.C_6^3:D_6$ ($C_2^2$), $C_2\times C_6^4.S_3^3$ ($C_2^2$), $C_6^4.D_6:S_3^2$ ($C_2^2$), $C_6^4.(C_{12}.S_3^2)$ ($C_2^2$), $C_6^2.C_6^4:D_6$ ($C_2^2$), $C_3^6.(C_2^3.\GL(2,\mathbb{Z}/4))$ ($C_2^2$), $C_6^4.(A_4\times S_3^2)$ ($C_2^2$), $C_6^2.C_6^4:C_{12}$ ($C_2^2$)

Order 373248: $C_3^6.C_2^6.C_2^3$ ($S_3$)

Order 279936: $C_6^4.S_3^3$ ($D_4$) x 8, $C_6^4.C_3^3.C_2^3$ ($D_4$) x 4, $C_3^6.C_2^4:C_3.C_2^3$ ($C_2^3$) x 3, $C_3^6.C_2^2\wr S_3$ ($C_2^3$) x 3, $C_6^4.C_3^3.C_2^3$ ($C_2^3$) x 2, $C_3^6.(C_2^4.S_4)$ ($C_2^3$) x 2, $C_3^6.C_2^5:C_{12}$ ($C_2^3$) x 2, $C_6^3.C_6^2:S_3^2$ ($C_2^3$), $C_6^4.(C_6\times S_3^2)$ ($C_2^3$), $C_6^3.C_6^3:C_6$ ($C_2^3$)

Order 186624: $C_3^6.C_2^6.C_2^2$ ($D_6$) x 6, $C_2^3\times C_3^6.C_2^5$ ($D_6$)

Order 139968: $(C_3^2\times C_6^4).C_6.C_2$ ($C_2\times D_4$) x 5, $C_6^4.C_3^3.C_2^2$ ($C_2\times D_4$) x 4, $C_6^4.C_3:S_3^2$ ($C_2\times D_4$) x 4, $C_6^4.C_3^2.D_6$ ($C_2\times D_4$) x 3, $C_6^4.C_3.C_6^2$ ($C_2\times D_4$) x 2, $(C_3^2\times C_6^4).C_6.C_2$ ($C_2^4$)

Order 93312: $C_3^6.C_2^6.C_2$ ($C_2\times D_6$) x 4, $C_2^2\times C_3^6.C_2^5$ ($C_2\times D_6$) x 2, $C_3^6.C_2^6.C_2$ ($S_4$), $C_2^3\times C_3^6.C_2^4$ ($C_2\times D_6$)

Order 69984: $C_6^4.C_3^3.C_2$ ($C_2^2\wr C_2$) x 4, $(C_3^2\times C_6^4).C_6$ ($C_2^2\times D_4$) x 2, $C_6^4.C_3^3.C_2$ ($C_2^2\times D_4$)

Order 46656: $C_3^6.C_2^5.C_2$ ($C_2\times S_4$) x 6, $C_2\times C_3^6.C_2^5$ ($S_3\times D_4$) x 4, $C_6^6$ ($S_3\times D_4$), $C_3^6.C_2^6$ ($C_2\times S_4$), $C_2^2\times C_3^6.C_2^4$ ($C_2^2\times D_6$), $C_2^2\times C_3^6.C_2^4$ ($S_3\times D_4$)

Order 34992: $C_6^4.C_3^3$ ($C_2^3:D_4$)

Order 31104: $C_6^4.C_6.C_2^2$ ($\SOPlus(4,2)$)

Order 23328: $C_3^6.C_2^5$ ($C_2^2\times S_4$) x 4, $C_3^6.C_2^4.C_2$ ($C_2^2\times S_4$) x 3, $C_2\times C_3^6.C_2^4$ ($D_4\times D_6$) x 2, $C_3\times C_6^5$ ($D_4\times D_6$)

Order 15552: $C_6^4.D_6$ ($S_3^2:C_2^2$) x 2, $C_6^4.C_6.C_2$ ($S_3^2:C_2^2$)

Order 11664: $C_3^6.C_2^4$ ($D_4\times S_4$) x 5, $C_3^6.C_2^4$ ($C_2^3:S_4$), $C_3^6.C_2^4$ ($C_2^3\times S_4$), $C_3^2\times C_6^4$ ($D_4\times S_4$), $C_3^2\times C_6^4$ ($C_2^4:D_6$)

Order 7776: $C_6^4.S_3$ ($D_6\wr C_2$) x 2, $C_6^4.C_6$ ($C_6^2:D_4$)

Order 5832: $C_3^6.C_2^3$ ($C_2^4:S_4$) x 3, $C_3^6.C_2^3$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 2, $C_3^3\times C_6^3$ ($\GL(2,\mathbb{Z}/4):C_2^2$)

Order 5184: $C_2^2\times C_6^4$ ($S_3^3:C_2$)

Order 3888: $C_6^4.C_3$ ($D_6^2:C_2^2$)

Order 2916: $C_3^5.D_6$ ($C_2\wr D_6$) x 5, $C_3^5.D_6$ ($C_2^5:S_4$), $C_3^4\times C_6^2$ ($C_2^6:D_6$), $C_3^4\times C_6^2$ ($C_2\wr D_6$)

Order 2592: $C_2\times C_6^4$ ($S_3^3:C_2^2$)

Order 1458: $C_3^5:S_3$ ($C_2^7:D_6$) x 2, $C_3^5\times C_6$ ($C_2^7:D_6$)

Order 1296: $C_6^4$ ($D_6^2:D_6$), $C_6^4$ ($D_6^2:D_6$)

Order 729: $C_3^6$ ($C_2^8:D_6$)

Order 648: $C_3\times C_6^3$ ($C_2\times D_6^2:D_6$)

Order 576: $C_2^4\times C_6^2$ ($C_3^4:(S_3\times D_4)$)

Order 324: $C_3^2\times C_6^2$ ($D_6\wr C_2\times S_4$), $C_3^2\times C_6^2$ ($D_6^2:(C_2\times S_4)$)

Order 288: $C_2^3\times C_6^2$ ($C_3^4:(D_4\times D_6)$)

Order 162: $C_3^3\times C_6$ ($D_6^2:(C_2^2\times S_4)$)

Order 144: $C_2^2\times C_6^2$ ($C_3^4:(D_4\times S_4)$), $C_2^2\times C_6^2$ ($C_6^2:S_3^2:D_6$)

Order 81: $C_3^4$ ($D_6^2.(D_4\times S_4)$)

Order 72: $C_2\times C_6^2$ ($C_3^4:\GL(2,\mathbb{Z}/4):C_2^2$)

Order 64: $C_2^6$ ($C_3^6.(S_3\times D_4)$)

Order 36: $C_6^2$ ($C_6^4.(S_3\times D_4)$), $C_6^2$ ($C_6^4.(S_3\times D_4)$)

Order 32: $C_2^5$ ($C_3^6.(D_4\times D_6)$)

Order 18: $C_3\times C_6$ ($C_6^4.(D_4\times D_6)$)

Order 16: $C_2^4$ ($C_3^6.(D_4\times S_4)$), $C_2^4$ ($C_3^4.D_6^2:D_6$)

Order 9: $C_3^2$ ($C_6^4.(D_4\times S_4)$)

Order 8: $C_2^3$ ($C_3^6.\GL(2,\mathbb{Z}/4):C_2^2$)

Order 4: $C_2^2$ ($C_6^4.S_3^3:C_2$), $C_2^2$ ($C_3^6.C_2\wr D_6$)

Order 2: $C_2$ ($C_6^4.S_3^3:C_2^2$)

Order 1: $C_1$ ($C_6^4.D_6^2:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $1986 \times 1986$ rational character table is not available for this group.