#27537935

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2186 ⋊ C₂
Sylow	C₅₃ ⋊ #33524

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	#33524
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2186

1118 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)(60,68)(65,79)(69,82)(70,83)(77,88)(80,86)(84,87)(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)(55,57)(58,60)(59,65)(61,69)(62,70)(63,68)(64,77)(66,80)(67,79)(71,84)(72,85)(73,83)(74,82)(75,86)(76,87)(78,88)(81,89);(58,59,63,67)(60,65,68,79)(61,72,74,81)(62,75,73,66)(64,76,78,71)(69,85,82,89)(70,86,83,80)(77,87,88,84);(54,55)(56,57)(59,64)(61,71)(62,73)(65,77)(66,81)(67,78)(69,84)(70,83)(72,75)(74,76)(79,88)(80,89)(82,87)(85,86)

Polycyclic presentation:

2^8,53|b^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,d^c=dh,i^c=i^52,i^d=i^52

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323411);