C₂×#20275256 #27386102

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#20275256

Splittings

Fitting	C₂×C₅₃×#342 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1913

Subgroups

	subgroup	G/H
center	C₂²	#16810501
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₅₃×#342	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#16811185	C₂
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×#342	D₅₃
hypocenter	C₅₃	C₂×#1913
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂×#342

2078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 71 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)(57,62)(58,64)(60,67)(61,68)(63,66)(65,70)(69,71);(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)(57,61,63,69)(59,64)(60,65)(62,68,66,71)(67,70);(56,57)(58,61,65,69)(59,66,67,62)(60,63)(64,68)(70,71);(54,55)

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171578);

C2×#20275256 #27386102