C₂×#20276354 #27536811

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

Aliases

direct product

C₂×#20276354

Splittings

Fitting	C₂×C₅₃×#410 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2120

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133131
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#410	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#16811130	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	C₂×#410	D₅₃
hypocenter	C₅₃	C₂×#2120
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#410

1398 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)(56,58,60,65)(57,62,63,69)(59,64,67,71)(61,68,66,70);(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)(57,61)(62,68)(63,66)(69,70);(56,57)(58,62)(59,66,67,61)(60,63)(64,70,71,68)(65,69);(54,55)

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322287);

C2×#20276354 #27536811