C₂×#20276442 #27537019

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276442

Splittings

Fitting	C₂²×#108×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2143

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×#13133239	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#2143
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 points:

(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)(58,68)(63,78)(67,83)(69,85)(70,77)(72,87)(79,80)(81,84);(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)(57,63)(59,72)(60,70)(61,77)(62,68)(64,81)(65,79)(66,78)(67,71)(69,74)(73,85)(75,87)(76,80)(82,84)(83,86);(56,57,60,65,62,66,61,76)(58,67,70,84,68,83,77,81)(59,71,73,82,75,86,74,64)(63,72,79,85,78,87,80,69);(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, 322495);

C2×#20276442 #27537019