C₂×#20276975 #27539490

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

Aliases

direct product

C₂×#20276975

Splittings

Fitting	C₂²×SD₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2249

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×SD₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×#13133158	C₂
hypercenter	C₂²×SD₃₂	D₅₃
hypocenter	C₅₃	C₂×#2249
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×SD₃₂

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

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324966);

C2×#20276975 #27539490