C₂×#20276313 #27536728

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#20276313

Splittings

Fitting	C₂×C₅₃×#346 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2108

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	C₂×#2108
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 91 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)(62,70)(66,76)(71,82)(72,83)(77,86)(78,87)(84,90)(88,91);(56,57,58,59)(60,62)(61,66)(63,71)(64,72)(65,70)(67,77)(68,78)(69,76)(73,84)(74,82)(75,83)(79,88)(80,86)(81,87)(85,90)(89,91);(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)(60,61,63,67,64,68,73,79,65,69,74,80,75,81,85,89)(62,66,71,77,72,78,84,88,70,76,82,86,83,87,90,91);(54,55)

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322204);

C2×#20276313 #27536728