C₂×#20276667 #27537983

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

Aliases

direct product

C₂×#20276667

Splittings

Fitting	C₂×C₅₃×#360 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2191

Subgroups

	subgroup	G/H
center	C₂²	#16810758
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#360	C₂
socle	C₂²×C₅₃	#447
prime radical	C₂×#16811420	C₂
hypercenter	C₂×#360	D₅₃
hypocenter	C₅₃	C₂×#2191
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

1128 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)(59,72)(63,82)(66,84)(67,77)(70,86)(71,87)(74,80)(81,85);(56,58)(57,62)(59,71)(60,76)(61,68)(63,81)(64,83)(65,78)(66,70)(67,74)(69,73)(72,87)(75,79)(77,80)(82,85)(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)(56,57,61,65)(58,66,68,84)(59,70,72,86)(60,75,73,64)(62,77,78,67)(63,80,82,74)(69,85,76,81)(71,83,87,79);(54,55)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323459);

C2×#20276667 #27537983