C₂²×#16810462 #27542212

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×#16810462

Splittings

Fitting	C₂²×C₃₂×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂²×SD₆₄

Subgroups

	subgroup	G/H
center	C₂³	D₈₄₈
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂²×C₃₂×C₅₃	C₂
socle	C₂³×C₅₃	D₁₆
prime radical	C₂²×D₈₄₈	C₂
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	C₂²×SD₆₄
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

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

normal subgroups only

This group has 502 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:

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

Conjugacy classes & character table

3404 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 327688);