#27538483

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2213 ⋊ C₂
Sylow	C₅₃ ⋊ #33594

Subgroups

	subgroup	G/H
center	C₂	#459×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2213	C₂
socle	C₂×C₅₃	C₂×#459
hypercenter	#2213	D₅₃
hypocenter	C₅₃	#33594
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2213

696 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)(54,58,61,59)(55,64,66,74)(56,60,69,68)(57,71,73,72)(62,65,75,82)(63,78,80,79)(67,76,84,83)(70,85,77,81);(54,57)(55,63)(56,67)(58,71)(59,72)(60,76)(61,73)(62,77)(64,78)(65,81)(66,80)(68,83)(69,84)(70,75)(74,79)(82,85);(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)(54,56,59,68,61,69,58,60)(55,62,64,65,66,75,74,82)(57,67,72,83,73,84,71,76)(63,77,78,81,80,70,79,85);(54,55)(56,65)(57,70)(58,64)(59,74)(60,75)(61,66)(62,68)(63,67)(69,82)(71,85)(72,81)(73,77)(76,78)(79,83)(80,84)

Polycyclic presentation:

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

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323959);