C₂×#20275215 #27385998

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

Aliases

direct product

C₂×#20275215

Splittings

Fitting	C₂×C₅₃×#339 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1906

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#339	C₂
socle	C₂²×C₅₃	#386
prime radical	C₂×#16811423	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#339	D₅₃
hypocenter	C₅₃	C₂×#1906
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#339

1798 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)(56,58)(57,62)(59,67)(60,65)(61,69)(63,70)(64,71)(66,68);(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)(57,61,63,71)(59,67)(60,65)(62,69,70,64)(66,68);(56,57)(58,61,65,71)(59,64)(60,63)(62,66,70,67)(68,69);(54,55)

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171474);

C2×#20275215 #27385998