C₂×#20275821 #27411275

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

Aliases

direct product

C₂×#20275821

Splittings

Fitting	C₂²×AGL(1,ℤ/8)×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2018

Subgroups

	subgroup	G/H
center	C₂³	#13133189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×AGL(1,ℤ/8)×C₅₃	C₂
socle	C₂³×C₅₃	#121
hypercenter	C₂²×AGL(1,ℤ/8)	D₅₃
hypocenter	C₅₃	C₂×#2018
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×AGL(1,ℤ/8)

2158 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)(61,67)(62,71)(63,64)(68,69)(70,74)(72,73);(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)(57,59)(60,62)(61,66)(63,72)(64,71)(65,73)(67,74)(68,70)(69,75);(56,57)(58,59)(60,61)(62,70)(63,68)(64,67)(65,69)(66,71)(72,75)(73,74);(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, 196751);

C2×#20275821 #27411275