C₂×#20272746 #27294394

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20272746

Splittings

Sylow

C₅₃ ⋊ C₂×#1354

Subgroups

	subgroup	G/H
center	C₂³	#13133131
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄⋊C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#324	C₂
socle	C₂³×C₅₃	#99
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₅₃	C₂³
hypercenter	C₂×#324	D₅₃
hypocenter	C₅₃	C₂×#1354
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ C₂×#324

652 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 123 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)(56,58,59,63)(57,60)(64,65)(66,70)(67,69)(68,71);(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)(56,57,58,62,59,60,63,61)(65,68,69,70);(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, 79870);