#27300164

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2150 ⋊ C₂
Sylow	C₅₃ ⋊ #12817

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#12817
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

430 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)(58,60)(59,65)(61,69)(62,75)(63,78)(64,79)(66,82)(67,83)(68,85)(70,88)(71,73)(72,89)(74,84)(76,87)(77,81)(80,86);(54,55,56,57)(59,64)(65,79)(66,80)(67,77)(68,84)(74,85)(81,83)(82,86);(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)(58,59)(60,68,69,84)(61,64)(62,74)(63,77)(65,70,79,87)(66,76)(67,73)(71,82,78,86)(72,85)(75,83,89,81)(80,88)

Polycyclic presentation:

2^8,53|a^2=f,b^2=dh,e^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,i^a=i^52,e^c=egh,f^c=fh,g^c=gh,i^c=i^52

Conjugacy classes & character table

1502 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85640);