#27537867

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2186 ⋊ C₂
Sylow	C₅₃ ⋊ #33516

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2186	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#2186	D₅₃
hypocenter	C₅₃	#33516
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2186

1150 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)(58,61,64,63)(59,66,68,78)(60,70,72,79)(62,73,77,76)(65,80,82,81)(67,74,84,88)(69,85,86,83)(71,75,87,89);(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)(55,57);(58,60)(59,65)(61,70)(62,75)(63,79)(64,72)(66,80)(67,83)(68,82)(69,74)(71,76)(73,87)(77,89)(78,81)(84,85)(86,88);(54,55)(56,57)(58,59)(60,69)(61,66)(62,74)(63,78)(64,68)(65,71)(67,76)(70,85)(72,86)(73,84)(75,80)(77,88)(79,83)(81,89)(82,87)

Polycyclic presentation:

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

Conjugacy classes & character table

812 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323343);