#27352562

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1540 ⋊ C₂
Sylow	C₅₃ ⋊ #17724

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1540	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	#20288584	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1540	D₅₃
hypocenter	C₅₃	#17724
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1540

1004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 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)(56,62)(60,65)(63,68)(67,69);(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)(54,56)(55,60)(57,63)(58,62)(59,65)(61,67)(64,68)(66,69)(70,73,75,74)(71,77,79,78)(72,81,83,82)(76,84,85,80);(55,59)(57,64)(60,65)(63,68)(70,72)(71,76)(73,81)(74,82)(75,83)(77,84)(78,80)(79,85);(54,55,57,61,58,59,64,66)(56,60,63,67,62,65,68,69)(70,71,74,78,75,79,73,77)(72,80,81,76,83,84,82,85)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,e^2=h,f^2=gh,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dg,i^c=i^52,i^d=i^52

Conjugacy classes & character table

1049 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 138038);