C₂×#20276269 #27413975

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20276269

Splittings

Fitting	C₂×C₅₃×H(3,ℤ/4) ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2098

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133214
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×H(3,ℤ/4)	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂²×#11719945	C₂²
hypercenter	C₂×H(3,ℤ/4)	D₅₃
hypocenter	C₅₃	C₂×#2098
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

1010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

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)(56,59)(57,64)(58,68)(60,74)(61,75)(62,77)(63,78)(65,81)(66,72)(67,84)(69,73)(70,86)(71,85)(76,82)(79,87)(80,83);(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)(56,58,62,69,61,71,60,70)(57,63,76,67,66,79,65,83)(59,73,77,85,75,86,74,68)(64,80,82,78,72,84,81,87);(56,57,60,65,61,66,62,76)(58,67,69,79,71,83,70,63)(59,72,74,82,75,64,77,81)(68,84,73,87,85,80,86,78);(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, 199451);

C2×#20276269 #27413975