#27435578

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2038 ⋊ C₂
Sylow	C₅₃ ⋊ #25442

Subgroups

	subgroup	G/H
center	C₂²	#16811255
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2038	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20284384	C₂
hypercenter	#2038	D₅₃
hypocenter	C₅₃	#25442
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2038

890 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)(58,61,63,77,64,78,62,76)(59,67,65,79,69,83,68,75)(60,72,71,85,74,87,73,86)(66,80,70,84,82,89,81,88);(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)(59,66)(61,72)(62,73)(63,71)(64,74)(65,70)(67,80)(68,81)(69,82)(75,88)(76,86)(77,85)(78,87)(79,84)(83,89);(54,55)(56,57)(59,65,69,68)(60,71,74,73)(67,79,83,75)(72,85,87,86);(58,59,62,65,64,69,63,68)(60,70,71,66,74,81,73,82)(61,75,76,67,78,79,77,83)(72,80,85,88,87,89,86,84)

Polycyclic presentation:

2^8,53|a^2=fh,b^2=ef,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fh,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, 221054);