C₂×#20282753 #27590230

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#20282753

Splittings

Sylow

C₅₃ ⋊ C₂×#2817

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#397	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂×#397	D₅₃
hypocenter	C₅₃	C₂×#2817
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#397

2478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 83 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)(62,66)(64,67)(68,72,73,70)(69,74,76,77)(71,78,81,80)(75,83,79,82);(57,59)(61,65)(64,67)(68,71,73,81)(69,75,76,79)(70,78,72,80)(74,82,77,83);(56,57)(58,59)(60,62)(61,64)(63,66)(65,67)(68,70,73,72)(69,74,76,77)(71,80,81,78)(75,83,79,82);(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)(60,61)(62,64)(63,65)(66,67)(68,69,73,76)(70,77,72,74)(71,79,81,75)(78,82,80,83);(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, 375706);

C2×#20282753 #27590230