#27299763

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#362 ⋊ C₂
Sylow	C₅₃ ⋊ #12738

Subgroups

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#362	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×#13133240	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#362	D₅₃
hypocenter	C₅₃	#12738
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 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)(55,57)(58,61,63,75)(59,65,68,67)(60,70,73,72)(62,74,78,87)(64,79,81,80)(66,82,84,76)(69,85,86,83)(71,77,89,88);(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)(58,60)(59,64)(61,70)(62,77)(63,73)(65,79)(66,83)(67,80)(68,81)(69,82)(71,87)(72,75)(74,89)(76,86)(78,88)(84,85);(54,55)(56,57)(58,59,61,65,63,68,75,67)(60,69,72,83,73,86,70,85)(62,76,74,66,78,82,87,84)(64,77,80,71,81,88,79,89)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dgh,f^b=fh,g^b=gh,i^b=i^52,i^c=i^52

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85239);