C₂×C₇×C₃₁⋊D₃₁ #41002069

order	26908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₂×C₃₁

Aliases

direct product	C₂×C₇×C₃₁⋊D₃₁
unambiguous semidirect product	(C₂×C₃₁)⋊(C₇×D₃₁)

Splittings

Fitting, socle	C₂×C₇×C₃₁² ⋊ C₂
derived, Sylow	C₃₁² ⋊ C₂²×C₇
Hall	C₇×C₃₁² ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇	C₃₁⋊D₃₁
commutator, cosocle, hypocenter	C₃₁²	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₃₁²	C₂
LCS	G ▷ C₃₁²
UCS	C₁ ◁ C₂×C₇

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 13454

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69);(39,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41)(70,71);(8,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10)

Conjugacy classes & character table

6748 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26908, 25);

C2×C7×C31⋊D31 #41002069