#27432107

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3063 ⋊ C₂
Sylow	C₅₃ ⋊ #25191

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3063	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20284016	C₂
hypercenter	#3063	D₅₃
hypocenter	C₅₃	#25191
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #3063

1010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|a^2=f,b^2=f,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,i^b=i^52,f^c=fh,g^c=gh,e^d=eh,i^d=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, 217583);