C₂×#20272664 #27293264

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20272664

Splittings

Sylow

C₅₃ ⋊ C₂×#1326

Subgroups

	subgroup	G/H
center	C₂³	#13133131
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#323	C₂
socle	C₂³×C₅₃	#99
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	C₂×#323	D₅₃
hypocenter	C₅₃	C₂×#1326
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ C₂×#323

770 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2)(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)(65,68,69,70)(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)(56,57,58,60,59,61,62,63)(64,65)(66,70,71,68)(67,69);(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, 78740);