C₂×#20276876 #27538724

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

Aliases

direct product

C₂×#20276876

Splittings

Sylow

C₅₃ ⋊ C₂×#2229

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×#2229
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

542 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)(64,66,67,75)(65,71,70,85)(68,81,78,95)(69,84,79,77)(72,90,86,94)(73,91,83,88)(74,93,92,80)(76,89,87,82);(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)(71,85)(72,86)(73,83)(74,92)(80,93)(88,91)(90,94);(56,57,59,61)(58,62,63,60)(64,65,67,70)(66,74,75,92)(68,80,78,93)(69,83,79,73)(71,87,85,76)(72,89,86,82)(77,94,84,90)(81,91,95,88);(54,55)

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324200);

C2×#20276876 #27538724