C₂×#27162406 #38760513

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

Aliases

direct product

C₂×#27162406

Splittings

Fitting, socle	C₂×C₇²×C₄₃ ⋊ C₂×C₃
derived, Hall	C₇²×C₄₃ ⋊ C₂²×C₃
Hall	C₂×C₄₃×C₇⋊D₇ ⋊ C₃
Hall	#20263479 ⋊ C₂²
Hall	C₂×C₇⋊D₇ ⋊ C₄₃⋊C₃
Sylow	C₇² ⋊ C₂²×C₄₃⋊C₃
Sylow	C₄₃ ⋊ C₂×#64424

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#27162406
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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25284, 320);

C2×#27162406 #38760513