C₂×#20274108 #27366447

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20274108

Splittings

Sylow

C₅₃ ⋊ C₂×#1674

Subgroups

	subgroup	G/H
center	C₂³	#13133190
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#332	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	C₂×#332	D₅₃
hypocenter	C₅₃	C₂×#1674
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 79 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)(64,66,68,74)(65,69,71,78)(67,76,77,73)(70,79,75,72);(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)(56,57,61,63,59,62,58,60);(57,60)(58,61)(62,63)(64,65,68,71)(66,72,74,79)(67,75,77,70)(69,76,78,73);(54,55)

Conjugacy classes & character table

1046 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 151923);

C2×#20274108 #27366447