#27510495

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1881 ⋊ C₂
Sylow	C₅₃ ⋊ #31281

Subgroups

	subgroup	G/H
center	C₄	#16811366
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1881	C₂
socle	C₂×C₅₃	#2684
prime radical	#20284061	C₂
hypercenter	#1881	D₅₃
hypocenter	C₅₃	#31281
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1881

744 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)(54,55)(56,66)(57,62)(58,64)(59,63)(60,65)(61,68)(67,79)(69,78)(70,77)(71,74)(72,82)(73,81)(75,80)(76,83)(84,85);(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)(54,57)(55,62)(56,67)(58,73)(59,74)(60,75)(61,76)(63,71)(64,81)(65,80)(66,79)(68,83)(69,72)(70,84)(77,85)(78,82);(54,56,59,69,60,70,58,68)(55,61,64,77,65,78,63,66)(57,72,73,67,75,83,74,84)(62,79,71,82,80,85,81,76);(54,55)(56,66)(57,71)(58,64)(59,63)(60,65)(61,68)(62,73)(67,82)(69,78)(70,77)(72,85)(74,80)(75,81)(76,84)(79,83)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 295971);