#27393265

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1915 ⋊ C₂
Sylow	C₅₃ ⋊ #21780

Subgroups

	subgroup	G/H
center	C₂	#20277289
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1915	C₂
socle	C₂×C₅₃	#2308
prime radical	#20288364	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1915	D₅₃
hypocenter	C₅₃	#21780
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1915

1236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

2^8,53|a^2=g,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,i^c=i^52,g^d=gh,i^d=i^52

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 178741);