C₂×#20276561 #27537352

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276561

Splittings

Fitting	C₂×C₅₃×#366 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×C₄×D₁₆

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₈₄₈	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 75 points:

(1,2)(3,4)(5,6)(7,8)(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)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(56,58,59,57);(2,3)(4,5)(6,7)(8,9)(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)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(61,65)(62,69)(63,70)(66,67)(68,74)(71,75)(72,73);(56,57,59,58)(60,61)(62,65)(63,72)(64,68)(66,69)(67,70)(71,74)(73,75);(54,55)

Conjugacy classes & character table

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322828);

C2×#20276561 #27537352