#27432195

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2070 ⋊ C₂
Sylow	C₅₃ ⋊ #25197

Subgroups

	subgroup	G/H
center	C₂	#20278246
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2070	C₂
socle	C₂×C₅₃	#2424
hypercenter	#2070	D₅₃
hypocenter	C₅₃	#25197
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2070

968 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)(54,55,60,64)(56,61,68,77)(57,62,71,79)(58,72,59,63)(65,67,76,73)(66,75,84,81)(69,74,78,70)(80,82,85,83);(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)(54,57)(55,62)(56,66)(58,70)(59,74)(60,71)(61,75)(63,78)(64,79)(65,80)(67,82)(68,84)(69,72)(73,83)(76,85)(77,81);(54,56,59,67,60,68,58,73)(55,61,63,76,64,77,72,65)(57,66,70,83,71,84,74,82)(62,75,69,80,79,81,78,85);(54,55,59,63,60,64,58,72)(56,65,67,61,68,76,73,77)(57,69,70,79,71,78,74,62)(66,81,83,85,84,75,82,80)

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 217671);