C₂×#27162432 #38760508

order	25284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#27162432

Splittings

Fitting, socle	C₂×C₇²×C₄₃ ⋊ C₂×C₃
derived, Hall	C₇²×C₄₃ ⋊ C₂²×C₃
Hall	C₂×C₇²×D₄₃ ⋊ C₃
Hall	#20263480 ⋊ C₂²
Hall	C₂×D₄₃ ⋊ #3861
Sylow	C₇² ⋊ C₂×D₄₃⋊C₃
Sylow	C₄₃ ⋊ C₂²×#3861

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#27162432
commutator, cosocle, hypocenter	C₇²×C₄₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₂

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 4214

A minimal faithful permutation representation — 3 generators on 59 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25284, 315);

C2×#27162432 #38760508