#27416627

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2031 ⋊ C₂
Sylow	C₅₃ ⋊ #23993

Subgroups

	subgroup	G/H
center	C₂	#20278035
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2031	C₂
socle	C₂×C₅₃	#2404
prime radical	#20284088	C₂
hypercenter	#2031	D₅₃
hypocenter	C₅₃	#23993
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2031

1152 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)(54,57)(55,62)(56,65)(58,68)(59,69)(60,67)(61,75)(63,77)(64,78)(66,79)(70,82)(71,81)(72,83)(73,80)(74,84)(76,85);(54,56)(55,61)(57,65)(58,66)(59,72)(60,73)(62,75)(63,74)(64,70)(67,80)(68,79)(69,83)(71,76)(77,84)(78,82)(81,85);(55,60)(56,64,66,72)(61,74,73,76)(62,67)(63,71)(65,78,79,83)(75,84,80,85)(77,81);(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)(54,55,58,60)(56,61,66,73)(57,67,68,62)(59,71,70,63)(64,74,72,76)(65,75,79,80)(69,77,82,81)(78,84,83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 202103);