#27302606

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #13262

Subgroups

	subgroup	G/H
center	C₂²	#16810817
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#362	C₂
socle	C₂²×C₅₃	#459
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#362	D₅₃
hypocenter	C₅₃	#13262
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

462 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 113 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) = 3392

A minimal faithful permutation representation — 3 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)(58,61,64,75,65,76,63,62)(59,67,78,85,69,84,68,74)(60,71,79,89,73,88,72,77)(66,80,82,87,83,86,81,70);(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)(54,56,57,55)(58,60,65,73)(59,66,69,83)(61,71,76,88)(62,77,75,89)(63,72,64,79)(67,80,84,86)(68,81,78,82)(70,85,87,74);(54,55,57,56)(58,59,62,67,63,78,76,85,65,69,75,84,64,68,61,74)(60,70,71,81,79,86,89,83,73,87,88,82,72,80,77,66)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dfgh,c^2=eh,d^2=gh,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,f^b=fg,g^b=gh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 88082);