#27435641

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2048 ⋊ C₂
Sylow	C₅₃ ⋊ #25447

Subgroups

	subgroup	G/H
center	C₂	#20278298
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2048	C₂
socle	C₂×C₅₃	#2428
hypercenter	#2048	D₅₃
hypocenter	C₅₃	#25447
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2048

892 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)(54,58,60,75,61,76,59,74)(55,63,72,84,57,71,64,73)(56,66,68,82,69,83,67,81)(62,77,79,85,70,80,65,78);(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)(55,61)(56,62)(58,71)(59,72)(60,64)(63,76)(65,67)(66,77)(68,79)(69,70)(73,75)(74,84)(78,81)(80,83)(82,85);(54,56)(55,62,57,70)(58,66)(59,67)(60,68)(61,69)(63,77,71,80)(64,65,72,79)(73,78,84,85)(74,81)(75,82)(76,83);(54,55,60,64,61,57,59,72)(56,65,68,70,69,79,67,62)(58,73,75,71,76,84,74,63)(66,80,82,85,83,77,81,78)

Polycyclic presentation:

2^8,53|a^2=f,b^2=ef,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,f^c=fh,i^c=i^52,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, 221117);