C₂×#20275970 #27411875

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20275970

Splittings

Fitting	C₂×C₅₃×#418 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2045

Subgroups

	subgroup	G/H
center	C₂³	#13133189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#418	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂³×#11719922	C₂
hypercenter	C₂×#418	D₅₃
hypocenter	C₅₃	C₂×#2045
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#418

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 3392

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)(57,59)(60,62,63,71,64,70,72,75)(61,66,67,74,68,69,65,73);(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)(62,70)(63,72)(66,74)(67,68)(69,73);(56,57)(58,59)(60,61)(62,69)(63,65)(64,68)(66,70)(67,72)(71,74)(73,75);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 197351);

C2×#20275970 #27411875