#27385689

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1886 ⋊ C₂
Sylow	C₅₃ ⋊ #20955

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1886	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂²×#11719949	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1886	D₅₃
hypocenter	C₅₃	#20955
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1886

1710 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)(58,62,64,63)(59,67,69,68)(60,72,74,73)(61,76,78,77)(65,70,81,80)(66,82,75,83)(71,85,86,87)(79,84,88,89);(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)(58,61)(59,66)(60,71)(62,76)(63,77)(64,78)(65,79)(67,82)(68,83)(69,75)(70,84)(72,85)(73,87)(74,86)(80,89)(81,88);(54,56,57,55)(58,60)(59,65)(61,71)(62,72)(63,73)(64,74)(66,79)(67,70)(68,80)(69,81)(75,88)(76,85)(77,87)(78,86)(82,84)(83,89);(54,55,57,56)(58,59,63,67,64,69,62,68)(60,70,72,65,74,80,73,81)(61,75,76,82,78,66,77,83)(71,84,87,88,86,89,85,79)

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 171165);