C₂×#20276530 #27537218

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20276530

Splittings

Fitting	C₂×C₅₃×#364 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2166

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	C₂×#2166
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 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)(57,63,64,66)(58,68)(59,72)(62,75,67,78)(65,82,80,79)(70,83)(73,77)(76,87,81,86);(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)(56,58)(57,62)(59,71)(60,70)(61,68)(63,78)(64,67)(65,81)(66,75)(69,72)(73,85)(74,83)(76,80)(77,84)(79,86)(82,87);(56,57,59,65,60,66,73,82,61,64,72,80,74,63,77,79)(58,67,69,81,70,78,84,87,68,62,71,76,83,75,85,86);(54,55)

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322694);

C2×#20276530 #27537218