C₂×#20276359 #27536817

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

Aliases

direct product

C₂×#20276359

Splittings

Fitting	C₂²×#99×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2121

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133131
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₂²×#99×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#16811182	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	C₂²×#99	D₅₃
hypocenter	C₅₃	C₂×#2121
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ C₂²×#99

1494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 71 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)(58,65)(61,69)(64,70)(66,71);(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)(57,61)(59,66)(60,65)(62,64)(63,69)(67,70)(68,71);(56,57)(58,64)(59,67,68,62)(60,63)(61,66,69,71)(65,70);(54,55)

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322293);

C2×#20276359 #27536817