C₂×#20272895 #27304520

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

Aliases

direct product

C₂×#20272895

Splittings

Sylow

C₅₃ ⋊ C₂×#1416

Subgroups

	subgroup	G/H
center	C₂³	#13133158
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle, prime radical	C₂³×C₅₃	D₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	C₂×#1416
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#366

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

normal subgroups only

This group has 302 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) = 91
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 3 generators on 91 points:

(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)(60,61,65,69)(62,71,74,88)(63,75,77,91)(64,78,76,68)(66,79,81,70)(67,83,84,72)(73,90,86,82)(80,87,85,89);(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)(56,57,58,59)(61,66)(62,70)(63,72)(64,76)(67,82)(68,85)(69,81)(71,75)(73,89)(74,79)(77,83)(78,80)(84,90)(86,87)(88,91);(54,55)

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 89996);