#27426550

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2064 ⋊ C₂
Sylow	C₅₃ ⋊ #24798

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2064	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133239	C₂²
hypercenter	#2064	D₅₃
hypocenter	C₅₃	#24798
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2064

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(58,61,75,82,63,77,62,76)(59,66,68,83,69,84,67,74)(60,70,72,87,73,88,71,86)(64,78,81,89,79,85,65,80);(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)(55,57)(58,60)(59,65)(61,70)(62,71)(63,73)(64,68)(66,80)(67,79)(69,81)(72,75)(74,85)(76,86)(77,88)(78,83)(82,87)(84,89);(58,60,62,71,63,73,75,72)(59,64,67,65,69,79,68,81)(61,70,76,86,77,88,82,87)(66,78,74,80,84,85,83,89);(54,55,56,57)(58,59,63,69)(60,65,73,81)(61,74,77,83)(62,68,75,67)(64,72,79,71)(66,82,84,76)(70,85,88,78)(80,87,89,86)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 212026);