C₂×#20276885 #27538821

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

Aliases

direct product

C₂×#20276885

Splittings

Fitting	C₂×C₅₃×#365 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2230

Subgroups

	subgroup	G/H
center	C₂²	#16810912
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂×#13133239	C₂²
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	C₂×#2230
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

558 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)(57,63)(59,67)(60,69)(61,70)(62,68)(64,71)(65,66);(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)(56,57,59,60,62,65,61,64)(58,63,67,69,68,66,70,71);(57,60)(58,66,68,63)(59,61)(64,65)(67,69,70,71);(54,55)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324297);

C2×#20276885 #27538821