#27435481

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2047 ⋊ C₂
Sylow	C₅₃ ⋊ #25435

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2047	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#11719945	C₂³
hypercenter	#2047	D₅₃
hypocenter	C₅₃	#25435
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2047

880 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)(58,61,63,75,64,76,62,74)(59,67,77,87,69,83,68,73)(60,70,78,88,72,86,71,85)(65,79,80,89,81,84,66,82);(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)(54,55)(56,57)(58,60)(59,66)(61,70)(62,71)(63,78)(64,72)(65,77)(67,82)(68,81)(69,80)(73,84)(74,85)(75,88)(76,86)(79,87)(83,89);(54,55,56,57)(58,60,63,78,64,72,62,71)(59,65,77,80,69,81,68,66)(61,70,75,88,76,86,74,85)(67,79,87,89,83,84,73,82);(54,55,56,57)(58,59,64,69)(60,66,72,80)(61,73,76,87)(62,77,63,68)(65,78,81,71)(67,75,83,74)(70,84,86,79)(82,88,89,85)

Polycyclic presentation:

2^8,53|a^2=fh,b^2=efh,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, 220957);