C₂×#27162414 #38760517

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

Aliases

direct product

C₂×#27162414

Splittings

Fitting, socle	C₂×C₇²×C₄₃ ⋊ C₂×C₃
derived, Hall	C₇²×C₄₃ ⋊ C₂²×C₃
Hall	C₂×C₇×D₃₀₁ ⋊ C₃
Hall	#20263479 ⋊ C₂²
Sylow	C₇² ⋊ C₂×D₄₃⋊C₃
Sylow	C₄₃ ⋊ C₂×#64423

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#27162414
commutator, cosocle, hypocenter	C₇²×C₄₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₂

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 1204

A minimal faithful permutation representation — 2 generators on 59 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25284, 324);

C2×#27162414 #38760517