C₂×#20289526 #28297989

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#20289526

Splittings

Fitting	C₂×C₅₃×#585 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×H(7,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²	C₂⁵×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂×C₅₃×#585	C₂
socle	C₂²×C₅₃	C₂⁶
hypercenter	C₂×#585	D₅₃
hypocenter	C₅₃	C₂×H(7,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×#585

60570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 7 generators on 71 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)(61,64)(62,69)(68,70);(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)(58,66)(61,64)(65,71)(68,70);(59,67)(62,69)(65,71)(68,70);(56,59)(57,62)(58,65)(60,67)(61,68)(63,69)(64,70)(66,71);(56,58)(57,61)(59,65)(60,66)(62,68)(63,64)(67,71)(69,70);(56,57)(58,64)(59,62)(60,63)(61,66)(65,70)(67,69)(68,71);(54,55)

Conjugacy classes & character table

1898 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1083465);

C2×#20289526 #28297989