C₂×#20276421 #27536973

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

Aliases

direct product

C₂×#20276421

Splittings

Fitting	C₂×C₅₃×#358 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2138

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#358	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×#13133240	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#358	D₅₃
hypocenter	C₅₃	C₂×#2138
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

1190 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)(60,63,66,65)(61,68,70,80)(62,72,74,81)(64,75,79,78)(67,82,84,83)(69,76,86,90)(71,87,88,85)(73,77,89,91);(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)(60,62)(61,67)(63,72)(64,77)(65,81)(66,74)(68,82)(69,85)(70,84)(71,76)(73,78)(75,89)(79,91)(80,83)(86,87)(88,90);(56,57,58,59)(60,61)(62,71)(63,68)(64,76)(65,80)(66,70)(67,73)(69,78)(72,87)(74,88)(75,86)(77,82)(79,90)(81,85)(83,91)(84,89);(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, 322449);

C2×#20276421 #27536973