C₂×#20274768 #27383753

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20274768

Splittings

Fitting	C₂×C₅₃×#344 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1814

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133190
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#344	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂²×#11719945	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#344	D₅₃
hypocenter	C₅₃	C₂×#1814
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#344

1330 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)(60,73)(61,76)(62,74)(63,77)(65,79)(66,80)(67,81)(69,83)(70,85)(71,84)(72,75)(78,87)(82,86);(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)(56,58,60,69,62,71,61,70)(57,63,65,67,66,78,75,82)(59,68,73,83,74,84,76,85)(64,77,79,81,80,87,72,86);(56,57)(58,67)(59,72,74,79)(60,75)(61,65)(62,66)(63,69)(64,73,80,76)(68,77,84,87)(70,78)(71,82)(81,83,86,85);(54,55)

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 169229);

C2×#20274768 #27383753