#27290729

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1742 ⋊ C₂
Sylow	C₅₃ ⋊ #10996

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1742	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288048	C₂
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1742	D₅₃
hypocenter	C₅₃	#10996
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

1306 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)(63,72)(64,76)(66,79)(67,78)(68,82)(70,83)(71,84)(74,86)(75,85)(77,87)(80,89)(81,88);(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)(54,55)(56,57)(58,60)(59,64)(61,69)(62,74)(63,70)(65,76)(66,68)(67,81)(71,75)(72,83)(73,86)(77,80)(78,88)(79,82)(84,85)(87,89);(58,59,62,67,63,66,75,80)(60,68,71,81,70,64,74,77)(61,65,73,78,72,79,85,89)(69,76,84,87,83,82,86,88)

Polycyclic presentation:

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

Conjugacy classes & character table

1196 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 76205);