C₂×#20276520 #27537165

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#20276520

Splittings

Fitting	C₂×C₄×C₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2164

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2164
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 6784

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

Conjugacy classes & character table

3440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322641);

C2×#20276520 #27537165