#27537502

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2179 ⋊ C₂
Sylow	C₅₃ ⋊ #33463

Subgroups

	subgroup	G/H
center	C₄	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2179	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	#20287848	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2179	D₅₃
hypocenter	C₅₃	#33463
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

652 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)(61,78)(62,80)(64,65)(66,83)(73,81)(77,85)(79,82);(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)(54,57,59,71,60,72,75,58)(55,62,64,81,63,80,65,73)(56,67,68,76,70,84,69,74)(61,77,79,83,78,85,82,66);(54,56)(55,61)(57,67)(58,74)(59,68)(60,70)(62,77)(63,78)(64,79)(65,82)(66,73)(69,75)(71,76)(72,84)(80,85)(81,83);(54,55,59,64,60,63,75,65)(56,66,68,77,70,83,69,85)(57,62,71,81,72,80,58,73)(61,76,79,84,78,74,82,67)

Polycyclic presentation:

2^8,53|a^2=f,c^2=f,e^2=gh,f^2=h,g^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gh,e^b=eg,g^b=gh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322978);