C₂×#20275612 #27388065

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#20275612

Splittings

Fitting	C₂×C₅₃×#421 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1976

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#421	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂²×#13133237	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#421	D₅₃
hypocenter	C₅₃	C₂×#1976
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#421

1474 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)(56,59,60,70,61,71,73,84)(57,63,65,69,64,77,72,83)(58,66,68,80,67,79,81,85)(62,74,75,78,76,86,82,87);(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)(57,62,64,76)(59,66)(60,68)(61,67)(63,74,77,86)(65,75,72,82)(69,78,83,87)(70,80)(71,79)(73,81)(84,85);(56,57)(58,62,67,76)(59,69)(60,72)(61,64)(63,70)(65,73)(66,78,79,87)(68,82,81,75)(71,83)(74,85,86,80)(77,84);(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, 173541);

C2×#20275612 #27388065