#27387642

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1952 ⋊ C₂
Sylow	C₅₃ ⋊ #21181

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1952	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂³×D₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1952	D₅₃
hypocenter	C₅₃	#21181
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1952

874 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)(65,70,80,83)(66,75,82,81)(71,86,88,87)(79,84,85,89);(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)(58,61,63,77,64,78,62,76)(59,66,67,75,69,82,68,81)(60,71,73,87,74,88,72,86)(65,79,70,84,80,85,83,89);(54,56,57,55)(58,60)(59,65)(61,71)(62,72)(63,73)(64,74)(66,79)(67,70)(68,83)(69,80)(75,84)(76,86)(77,87)(78,88)(81,89)(82,85);(54,55,57,56)(58,59)(60,70)(61,75)(62,67)(63,68)(64,69)(65,73)(66,77)(71,85)(72,80)(74,83)(76,82)(78,81)(79,88)(84,87)(86,89)

Polycyclic presentation:

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

Conjugacy classes & character table

896 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 173118);