#27538840

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2231 ⋊ C₂
Sylow	C₅₃ ⋊ #33638

Subgroups

	subgroup	G/H
center	C₂	#476×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2231	C₂
socle	C₂×C₅₃	C₂×#476
prime radical	#135×D₅₃	C₂²
hypercenter	#2231	D₅₃
hypocenter	C₅₃	#33638
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2231

514 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)(55,63)(61,65)(62,77)(64,79)(74,82)(75,81)(76,80)(78,85);(54,56)(55,62)(57,67)(58,71)(59,73)(60,68)(61,74)(63,77)(64,80)(65,82)(66,69)(70,84)(72,83)(75,78)(76,79)(81,85);(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)(54,55,57,61,60,63,59,65)(56,62,67,74,68,77,73,82)(58,64,69,75,72,79,70,81)(66,78,83,76,84,85,71,80);(55,61)(56,66,68,84)(57,59)(58,70)(62,76,77,80)(63,65)(67,71,73,83)(69,72)(74,78,82,85)(75,81)

Polycyclic presentation:

2^8,53|a^2=eg,b^2=e,e^2=h,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fg,g^a=gh,d^b=dh,i^b=i^52,f^c=fgh,g^c=gh,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, 324316);