Abstract group 559872.bu: $C_3^5.C_2\wr S_3^2$

• Label: 559872.bu
• Order: \( 2^{8} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^5.C_2\wr S_3^2$
Order:	\(559872\)\(\medspace = 2^{8} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_2\times C_6^3).C_3^2.C_6^3.C_2^4$, of order \(13436928\)\(\medspace = 2^{11} \cdot 3^{8} \)
Composition factors:	$C_2$ x 8, $C_3$ x 7
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	2655	6722	12960	187614	12960	266976	69984	559872
Conjugacy classes	1	14	30	13	706	8	190	96	1058
Divisions	1	14	23	13	408	5	105	49	618
Autjugacy classes	1	11	12	7	163	2	33	10	239

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=c^{6}=d^{6}=e^{6}=f^{6}=g^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,4,10)(2,6,9,8,15,16,11,17,3)(5,12,7)(13,14,18)(20,24,26,25,30,21), (1,2) \!\cdots\! \rangle$
Transitive group:	36T30418				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_6^5$ . $(S_3\times D_6)$ (2)	$(C_2\times C_6^5)$ . $S_3^2$ (2)	$(C_6^4.C_6^2)$ . $D_6$ (4)	$C_6^4$ . $(C_6^2:D_6)$	all 60

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

Homology

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

Subgroups

There are 111 normal subgroups (51 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^2.C_2^4:\He_3.D_6.C_2$
Commutator:	$G' \simeq$ $C_6^4.C_3^3.C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^3\times C_6$	$G/\Phi \simeq$ $C_2\times A_4^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^5$	$G/\operatorname{Fit} \simeq$ $S_3^2$
Radical:	$R \simeq$ $C_3^5.C_2\wr S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_3^3:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^3$

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 9 (order at least 62208) are shown, as well as all normal subgroups of any index.

Order 559872: $C_3^5.C_2\wr S_3^2$

Order 279936: $C_6^4.C_3^2:D_{12}$ x 2, $C_6^4.C_6^2:C_6$ x 2, $C_6^4.C_3^3.C_2^3$, $C_6^4.C_6:S_3^2$, $C_2^5.C_3^5:S_3^2$

Order 186624: $C_6^4.(S_3\times S_4)$ x 2, $C_3^4.C_2\wr S_3^2$

Order 139968: $C_6^4.C_3^3.C_2^2$ x 4, $C_3^5:\POPlus(4,3)$ x 2, $C_6^4.C_3^2:D_6$ x 2, $C_6^4.C_3^2:C_{12}$ x 2, $C_6^4.C_3^3.C_2^2$

Order 93312: $C_6^4.C_6^2.C_2$ x 2, $C_6^4:D_6:S_3$ x 2, $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_3.C_6^4:D_{12}$ x 2, $C_2\times C_6^4.S_3^2$ x 2, $C_6.C_6^4:D_6$ x 2, $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_6^4.C_6\wr C_2$ x 2, $C_6^4.C_6\wr C_2$ x 2, $C_6^4.(C_6\times A_4)$ x 2, $C_6^2.(C_6\times \He_3).C_2^4$, $C_6^4.C_6^2.C_2$, $C_6^4.(S_3\times D_6)$, $C_2.C_6^4:S_3^2$

Order 69984: $C_6^4.C_3^3.C_2$ x 3, $C_6^4.C_3^3.C_2$ x 2, $C_3^5.A_4\wr C_2$ x 2, $C_6^2.C_3^4.C_6.C_2^2$

Order 62208: $C_6^3.(D_6\times S_4)$ x 6, $(C_2^3\times C_6^3):S_3^2$ x 6, $C_6^3.(D_6\times S_4)$ x 2, $C_6^5.C_2^3$

Order 46656: $C_6^4.C_3.D_6$ x 2, $C_6^4.C_6^2$ x 2, $(C_3^2\times C_6^3).S_4$ x 2, $(C_3^2\times C_6^3).S_4$ x 2, $C_6^4.C_6^2$

Order 34992: $C_6^4.C_3^3$, $(C_3^3\times C_6^2):S_3^2$

Order 23328: $C_6^4.C_3^2.C_2$ x 2, $C_6^4.C_3.S_3$ x 2, $C_6^4.C_3.C_6$

Order 15552: $C_6^4.C_6.C_2$ x 2, $C_2\times C_6^5$

Order 11664: $C_6^4.C_3^2$ x 2, $C_6^4.C_3^2$

Order 7776: $C_6^4.C_6$ x 2, $C_6^5$

Order 5184: $(C_2\times C_6^3).C_6.C_2$ x 6, $C_2^2\times C_6^4$

Order 3888: $C_6^4.C_3$ x 2, $C_3\times C_6^4$

Order 2592: $(C_2\times C_6^3).C_6$ x 6, $C_2\times C_6^4$

Order 2187: $C_3^4.C_3^3$

Order 1728: $C_2^3\times C_6^3$ x 2

Order 1296: $C_3\times C_6^2:A_4$ x 6, $C_6^4$

Order 972: $C_3^3\times C_6^2$

Order 864: $C_2^2\times C_6^3$ x 2

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

Order 486: $C_3^4\times C_6$

Order 432: $C_2\times C_6^3$ x 2

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

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

Order 256: $C_2^5:D_4$

Order 243: $C_3^5$

Order 192: $C_2^5\times C_6$ x 2

Order 162: $C_3^3\times C_6$

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

Order 108: $C_3\times C_6^2$ x 2

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 64: $C_2^6$

Order 54: $C_3^2\times C_6$ x 2

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

Order 36: $C_6^2$

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 18: $C_3\times C_6$

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 9 (order at least 62208) are shown, as well as all normal subgroups of any index.

Order 559872: $C_3^5.C_2\wr S_3^2$

Order 279936: $C_6^4.C_3^3.C_2^3$, $C_6^4.C_3^2:D_{12}$, $C_6^4.C_6:S_3^2$, $C_2^5.C_3^5:S_3^2$, $C_6^4.C_6^2:C_6$

Order 186624: $C_3^4.C_2\wr S_3^2$, $C_6^4.(S_3\times S_4)$

Order 139968: $C_6^4.C_3^3.C_2^2$ x 3, $C_6^4.C_3^3.C_2^2$, $C_3^5:\POPlus(4,3)$, $C_6^4.C_3^2:D_6$, $C_6^4.C_3^2:C_{12}$

Order 93312: $C_6^4.C_6^2.C_2$, $C_6^2.(C_6\times \He_3).C_2^4$, $C_6^4.C_6^2.C_2$, $C_6^4:D_6:S_3$, $C_6^4.(S_3\times D_6)$, $C_2.C_6^4:S_3^2$, $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$, $C_3.C_6^4:D_{12}$, $C_2\times C_6^4.S_3^2$, $C_6.C_6^4:D_6$, $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$, $C_6^4.C_6\wr C_2$, $C_6^4.C_6\wr C_2$, $C_6^4.(C_6\times A_4)$

Order 69984: $C_6^4.C_3^3.C_2$ x 2, $C_6^4.C_3^3.C_2$ x 2, $C_6^2.C_3^4.C_6.C_2^2$, $C_3^5.A_4\wr C_2$

Order 62208: $C_6^5.C_2^3$, $C_6^3.(D_6\times S_4)$, $C_6^3.(D_6\times S_4)$, $(C_2^3\times C_6^3):S_3^2$

Order 46656: $C_6^4.C_3.D_6$ x 2, $C_6^4.C_6^2$, $C_6^4.C_6^2$, $(C_3^2\times C_6^3).S_4$, $(C_3^2\times C_6^3).S_4$

Order 34992: $C_6^4.C_3^3$, $(C_3^3\times C_6^2):S_3^2$

Order 23328: $C_6^4.C_3^2.C_2$, $C_6^4.C_3.S_3$, $C_6^4.C_3.C_6$

Order 15552: $C_6^4.C_6.C_2$, $C_2\times C_6^5$

Order 11664: $C_6^4.C_3^2$, $C_6^4.C_3^2$

Order 7776: $C_6^5$, $C_6^4.C_6$

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

Order 3888: $C_3\times C_6^4$, $C_6^4.C_3$

Order 2592: $(C_2\times C_6^3).C_6$, $C_2\times C_6^4$

Order 2187: $C_3^4.C_3^3$

Order 1728: $C_2^3\times C_6^3$

Order 1296: $C_6^4$, $C_3\times C_6^2:A_4$

Order 972: $C_3^3\times C_6^2$

Order 864: $C_2^2\times C_6^3$

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

Order 486: $C_3^4\times C_6$

Order 432: $C_2\times C_6^3$

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

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

Order 256: $C_2^5:D_4$

Order 243: $C_3^5$

Order 192: $C_2^5\times C_6$ x 2

Order 162: $C_3^3\times C_6$

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

Order 108: $C_3\times C_6^2$

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 64: $C_2^6$

Order 54: $C_3^2\times C_6$

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

Order 36: $C_6^2$

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 18: $C_3\times C_6$

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 6: $C_6$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 559872: $C_3^5.C_2\wr S_3^2$ ($C_1$)

Order 279936: $C_6^4.C_3^2:D_{12}$ ($C_2$) x 2, $C_6^4.C_6^2:C_6$ ($C_2$) x 2, $C_6^4.C_3^3.C_2^3$ ($C_2$), $C_6^4.C_6:S_3^2$ ($C_2$), $C_2^5.C_3^5:S_3^2$ ($C_2$)

Order 139968: $C_6^4.C_3^3.C_2^2$ ($C_2^2$) x 2, $C_6^4.C_3^2:D_6$ ($C_2^2$) x 2, $C_6^4.C_3^2:C_{12}$ ($C_2^2$) x 2, $C_6^4.C_3^3.C_2^2$ ($C_2^2$)

Order 93312: $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$ ($S_3$) x 2, $C_6^4.C_6^2.C_2$ ($S_3$)

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

Order 46656: $C_6^4.C_3.D_6$ ($D_6$) x 2, $C_6^4.C_6^2$ ($D_6$) x 2, $(C_3^2\times C_6^3).S_4$ ($D_6$) x 2, $(C_3^2\times C_6^3).S_4$ ($D_6$) x 2, $C_6^4.C_6^2$ ($D_6$)

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

Order 23328: $C_6^4.C_3^2.C_2$ ($C_2\times D_6$) x 2, $C_6^4.C_3.S_3$ ($C_3:D_4$) x 2, $C_6^4.C_3.C_6$ ($C_2\times D_6$)

Order 15552: $C_6^4.C_6.C_2$ ($S_3^2$) x 2, $C_2\times C_6^5$ ($S_3^2$)

Order 11664: $C_6^4.C_3^2$ ($S_3\times D_4$) x 2, $C_6^4.C_3^2$ ($C_6:D_4$)

Order 7776: $C_6^4.C_6$ ($S_3\times D_6$) x 2, $C_6^5$ ($S_3\times D_6$)

Order 5184: $(C_2\times C_6^3).C_6.C_2$ ($C_3^2:D_6$) x 6, $C_2^2\times C_6^4$ ($C_3:S_3^2$)

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

Order 2592: $(C_2\times C_6^3).C_6$ ($C_6.S_3^2$) x 6, $C_2\times C_6^4$ ($C_6:S_3^2$)

Order 1728: $C_2^3\times C_6^3$ ($C_3^2:S_3^2$) x 2

Order 1296: $C_3\times C_6^2:A_4$ ($C_6^2:D_6$) x 6, $C_6^4$ ($C_6^2:D_6$)

Order 972: $C_3^3\times C_6^2$ ($\POPlus(4,3)$)

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

Order 576: $C_2^4\times C_6^2$ ($C_3^3:S_3^2$)

Order 486: $C_3^4\times C_6$ ($A_4^2:C_2^3$)

Order 432: $C_2\times C_6^3$ ($C_6^2:S_3^2$) x 2

Order 324: $C_3^2\times C_6^2$ ($A_4^2:D_6$)

Order 288: $C_2^3\times C_6^2$ ($C_2\times C_3^3:S_3^2$)

Order 243: $C_3^5$ ($C_2\wr S_3^2$)

Order 192: $C_2^5\times C_6$ ($C_3^4:S_3^2$), $C_2^5\times C_6$ ($C_3^4:S_3^2$)

Order 162: $C_3^3\times C_6$ ($C_2\times A_4^2:D_6$)

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

Order 108: $C_3\times C_6^2$ ($(C_3\times A_4^2):D_6$) x 2

Order 96: $C_2^4\times C_6$ ($C_3^4:C_3.D_6.C_2$), $C_2^4\times C_6$ ($C_3^3.C_3^3.C_2^3$)

Order 81: $C_3^4$ ($(C_2^2\times A_4^2):D_6$)

Order 64: $C_2^6$ ($C_3^5:S_3^2$)

Order 54: $C_3^2\times C_6$ ($C_3.(C_3\times A_4^2).C_2^3$) x 2

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

Order 36: $C_6^2$ ($(C_2\times C_6^3):S_3^2$)

Order 32: $C_2^5$ ($C_2\times C_3^5:S_3^2$)

Order 27: $C_3^3$ ($C_2^4:\He_3.D_6.C_2^2$) x 2

Order 18: $C_3\times C_6$ ($(C_2^2\times C_6^3):S_3^2$)

Order 16: $C_2^4$ ($(C_3^3\times C_6^2):S_3^2$)

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

Order 9: $C_3^2$ ($(C_2^3\times C_6^3):S_3^2$)

Order 6: $C_6$ ($C_6^4.C_3^2.C_2^3$), $C_6$ ($C_3^2.C_2^4:\He_3.D_6.C_2$)

Order 4: $C_2^2$ ($C_3^5:\POPlus(4,3)$)

Order 3: $C_3$ ($C_6^4.D_6:D_6$), $C_3$ ($C_3^3.A_4^2.C_6.C_2^3$)

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

Order 1: $C_1$ ($C_3^5.C_2\wr S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 559872: $C_3^5.C_2\wr S_3^2$ ($C_1$)

Order 279936: $C_6^4.C_3^3.C_2^3$ ($C_2$), $C_6^4.C_3^2:D_{12}$ ($C_2$), $C_6^4.C_6:S_3^2$ ($C_2$), $C_2^5.C_3^5:S_3^2$ ($C_2$), $C_6^4.C_6^2:C_6$ ($C_2$)

Order 139968: $C_6^4.C_3^3.C_2^2$ ($C_2^2$) x 2, $C_6^4.C_3^3.C_2^2$ ($C_2^2$), $C_6^4.C_3^2:D_6$ ($C_2^2$), $C_6^4.C_3^2:C_{12}$ ($C_2^2$)

Order 93312: $C_6^4.C_6^2.C_2$ ($S_3$), $(C_3^3\times C_6^2).\GL(2,\mathbb{Z}/4)$ ($S_3$)

Order 69984: $C_6^4.C_3^3.C_2$ ($D_4$), $C_6^4.C_3^3.C_2$ ($C_2^3$)

Order 46656: $C_6^4.C_3.D_6$ ($D_6$) x 2, $C_6^4.C_6^2$ ($D_6$), $C_6^4.C_6^2$ ($D_6$), $(C_3^2\times C_6^3).S_4$ ($D_6$), $(C_3^2\times C_6^3).S_4$ ($D_6$)

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

Order 23328: $C_6^4.C_3^2.C_2$ ($C_2\times D_6$), $C_6^4.C_3.S_3$ ($C_3:D_4$), $C_6^4.C_3.C_6$ ($C_2\times D_6$)

Order 15552: $C_6^4.C_6.C_2$ ($S_3^2$), $C_2\times C_6^5$ ($S_3^2$)

Order 11664: $C_6^4.C_3^2$ ($S_3\times D_4$), $C_6^4.C_3^2$ ($C_6:D_4$)

Order 7776: $C_6^5$ ($S_3\times D_6$), $C_6^4.C_6$ ($S_3\times D_6$)

Order 5184: $(C_2\times C_6^3).C_6.C_2$ ($C_3^2:D_6$), $C_2^2\times C_6^4$ ($C_3:S_3^2$)

Order 3888: $C_3\times C_6^4$ ($D_6:D_6$), $C_6^4.C_3$ ($D_6:D_6$)

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

Order 1728: $C_2^3\times C_6^3$ ($C_3^2:S_3^2$)

Order 1296: $C_6^4$ ($C_6^2:D_6$), $C_3\times C_6^2:A_4$ ($C_6^2:D_6$)

Order 972: $C_3^3\times C_6^2$ ($\POPlus(4,3)$)

Order 864: $C_2^2\times C_6^3$ ($C_2\times C_3^2:S_3^2$)

Order 576: $C_2^4\times C_6^2$ ($C_3^3:S_3^2$)

Order 486: $C_3^4\times C_6$ ($A_4^2:C_2^3$)

Order 432: $C_2\times C_6^3$ ($C_6^2:S_3^2$)

Order 324: $C_3^2\times C_6^2$ ($A_4^2:D_6$)

Order 288: $C_2^3\times C_6^2$ ($C_2\times C_3^3:S_3^2$)

Order 243: $C_3^5$ ($C_2\wr S_3^2$)

Order 192: $C_2^5\times C_6$ ($C_3^4:S_3^2$), $C_2^5\times C_6$ ($C_3^4:S_3^2$)

Order 162: $C_3^3\times C_6$ ($C_2\times A_4^2:D_6$)

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

Order 108: $C_3\times C_6^2$ ($(C_3\times A_4^2):D_6$)

Order 96: $C_2^4\times C_6$ ($C_3^4:C_3.D_6.C_2$), $C_2^4\times C_6$ ($C_3^3.C_3^3.C_2^3$)

Order 81: $C_3^4$ ($(C_2^2\times A_4^2):D_6$)

Order 64: $C_2^6$ ($C_3^5:S_3^2$)

Order 54: $C_3^2\times C_6$ ($C_3.(C_3\times A_4^2).C_2^3$)

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

Order 36: $C_6^2$ ($(C_2\times C_6^3):S_3^2$)

Order 32: $C_2^5$ ($C_2\times C_3^5:S_3^2$)

Order 27: $C_3^3$ ($C_2^4:\He_3.D_6.C_2^2$)

Order 18: $C_3\times C_6$ ($(C_2^2\times C_6^3):S_3^2$)

Order 16: $C_2^4$ ($(C_3^3\times C_6^2):S_3^2$)

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

Order 9: $C_3^2$ ($(C_2^3\times C_6^3):S_3^2$)

Order 6: $C_6$ ($C_6^4.C_3^2.C_2^3$), $C_6$ ($C_3^2.C_2^4:\He_3.D_6.C_2$)

Order 4: $C_2^2$ ($C_3^5:\POPlus(4,3)$)

Order 3: $C_3$ ($C_6^4.D_6:D_6$), $C_3$ ($C_3^3.A_4^2.C_6.C_2^3$)

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

Order 1: $C_1$ ($C_3^5.C_2\wr S_3^2$)

Series

Derived series

$C_3^5.C_2\wr S_3^2$

$\rhd$

$C_6^4.C_3^3.C_2$

$\rhd$

$C_6^4$

$\rhd$

$C_1$

Chief series

$C_3^5.C_2\wr S_3^2$

$\rhd$

$C_6^4.C_3^3.C_2^3$

$\rhd$

$C_6^4.C_3^3.C_2^2$

$\rhd$

$C_6^4.C_3^3.C_2$

$\rhd$

$C_6^4.C_3^3$

$\rhd$

$C_6^4.C_3^2$

$\rhd$

$C_3\times C_6^4$

$\rhd$

$C_6^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^5.C_2\wr S_3^2$

$\rhd$

$C_6^4.C_3^3.C_2$

$\rhd$

$C_6^4.C_3^3$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_2\times C_6$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

The $1058 \times 1058$ character table is not available for this group.

Rational character table

The $618 \times 618$ rational character table is not available for this group.