C₂²×#16810458 #27542207

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#16810458

Splittings

Fitting	C₂²×D₁₆×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×D₃₂

Subgroups

	subgroup	G/H
center	C₂³	#13133158
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂²×D₁₆×C₅₃	C₂
socle	C₂³×C₅₃	D₁₆
hypercenter	C₂²×D₁₆	D₅₃
hypocenter	C₅₃	C₂²×D₃₂
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×D₁₆

542 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 171 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) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 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)(59,64)(60,69)(61,70)(62,74)(65,66)(67,83)(68,79)(71,88)(72,86)(73,77)(75,87)(76,78)(80,89)(81,82)(84,85);(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)(58,59)(60,64)(61,73)(62,76)(63,68)(65,69)(66,70)(67,74)(71,83)(72,79)(75,89)(77,85)(78,84)(80,88)(81,86)(82,87);(56,57);(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, 327683);