#27290672

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1733 ⋊ C₂
Sylow	C₅₃ ⋊ #10985

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1733	C₂
socle	C₂²×C₅₃	#379
prime radical	C₂²×#11719945	C₂²
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1733	D₅₃
hypocenter	C₅₃	#10985
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1733

1270 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)(59,65)(60,69)(62,73)(63,74)(64,78)(66,80)(67,72)(68,82)(70,84)(71,85)(75,87)(76,88)(77,89)(79,86)(81,83);(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)(54,55)(56,57)(58,60,63,71)(59,64,67,79)(61,69,74,85)(62,76,77,70)(65,78,72,86)(66,81,75,68)(73,88,89,84)(80,83,87,82);(58,59)(60,68)(61,72,74,65)(62,75)(63,67)(64,70)(66,77)(69,83,85,82)(71,81)(73,80,89,87)(76,79)(78,84,86,88)

Polycyclic presentation:

2^8,53|a^2=g,b^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,i^b=i^52,g^c=gh,i^c=i^52

Conjugacy classes & character table

1190 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76148);