C₂×#20276991 #27539571

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

Aliases

direct product

C₂×#20276991

Splittings

Sylow

C₅₃ ⋊ C₂×#2253

Subgroups

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	C₂×#2253
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 95 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)(64,66,75,91,69,78,79,85,67,76,87,95,80,92,68,77)(65,71,81,90,73,86,84,88,70,83,94,93,82,89,72,74);(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)(56,58,59,63)(57,60,61,62)(65,70)(71,83)(72,84)(73,82)(74,88)(81,94)(86,89)(90,93);(56,57,59,61)(58,62,63,60)(64,65,67,70)(66,74,76,88)(68,81,79,94)(69,82,80,73)(71,85,83,77)(72,87,84,75)(78,93,92,90)(86,91,89,95);(54,55)

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 325047);

C2×#20276991 #27539571