C₂×#20275760 #27411050

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#20275760

Splittings

Fitting	C₂²×#137×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2007

Subgroups

	subgroup	G/H
center	C₂³	#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#137×C₅₃	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂×#16811201	C₂
hypercenter	C₂²×#137	D₅₃
hypocenter	C₅₃	C₂×#2007
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#137

2038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 75 points:

(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)(57,59)(60,62)(61,66)(63,71)(64,73)(65,72)(67,70)(68,75)(69,74);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(56,57)(58,59)(60,62)(61,66)(63,73)(64,71)(65,72)(67,75)(68,70)(69,74);(60,61,65,69)(62,70,72,75)(63,68,64,67)(66,73,74,71);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196526);

C2×#20275760 #27411050