#27422415

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3069 ⋊ C₂
Sylow	C₅₃ ⋊ #24475

Subgroups

	subgroup	G/H
center	C₂	#20278086
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3069	C₂
socle	C₂×C₅₃	#2410
prime radical	#20289492	C₂
hypercenter	#3069	D₅₃
hypocenter	C₅₃	#24475
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3069

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 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)(54,58,61,57)(55,64,67,63)(56,70,73,75)(59,76,60,77)(62,80,81,82)(65,74,66,78)(68,79,83,85)(69,71,84,72);(54,57,61,58)(55,63,67,64)(56,69,73,84)(59,76,60,77)(62,79,81,85)(65,74,66,78)(68,80,83,82)(70,71,75,72);(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)(55,62)(57,75)(58,70)(59,72)(60,71)(61,73)(63,82)(64,80)(65,83)(66,68)(67,81)(69,76)(74,85)(77,84)(78,79);(54,55,59,66,61,67,60,65)(56,68,72,81,73,83,71,62)(57,74,77,64,58,78,76,63)(69,79,75,82,84,85,70,80)

Polycyclic presentation:

2^8,53|a^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,g^b=gh,i^b=i^52,f^c=fh,g^c=gh,e^d=eh,i^d=i^52

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 207891);