C₂×#20272776 #27295456

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#20272776

Splittings

Sylow

C₅₃ ⋊ C₂×#1366

Subgroups

	subgroup	G/H
center	C₂²	#16810372
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle	C₂²×C₅₃	#333
prime radical	C₂³×C₅₃	#100
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	C₂×#1366
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 87 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)(56,58,61,68)(57,62)(59,71,74,76)(60,75,72,70)(63,67)(64,80)(65,84)(66,78)(69,73,85,83)(77,81)(79,86)(82,87);(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)(56,57,59,64,72,65,83,81,61,66,74,82,60,63,73,79)(58,67,69,80,75,78,71,86,68,84,85,87,70,62,76,77);(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, 80932);

C2×#20272776 #27295456