C₂×#20274364 #27371843

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#20274364

Splittings

Sylow

C₅₃ ⋊ C₂×#1718

Subgroups

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#487	C₂
socle, prime radical	C₂³×C₅₃	#127
hypercenter	C₂×#487	D₅₃
hypocenter	C₅₃	C₂×#1718
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 87 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)(58,65)(61,70)(64,71)(66,69)(72,74,76,82,75,81,83,87)(73,78,79,80,77,85,84,86);(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)(56,58,59,66,60,65,68,69)(57,61,62,64,63,70,67,71)(73,77)(78,85)(79,84)(80,86);(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)(56,57,60,63)(58,64,65,71)(59,67,68,62)(61,69,70,66)(72,73)(74,80)(75,77)(76,84)(78,82)(79,83)(81,86)(85,87);(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, 157319);

C2×#20274364 #27371843