#27393304

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1910 ⋊ C₂
Sylow	C₅₃ ⋊ #21785

Subgroups

	subgroup	G/H
center	C₂	#20277289
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1910	C₂
socle	C₂×C₅₃	#2308
prime radical	#20288507	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1910	D₅₃
hypocenter	C₅₃	#21785
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1910

1262 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)(54,58,60,57)(55,63,64,72)(56,68,70,67)(59,73,75,71)(61,77,66,83)(62,65,80,81)(69,85,74,84)(76,78,79,82);(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)(54,57,60,58)(55,62,64,80)(56,67,70,68)(59,73,75,71)(61,76,66,79)(63,65,72,81)(69,85,74,84)(77,78,83,82);(54,56)(55,61)(57,67)(58,68)(59,74)(60,70)(62,79)(63,77)(64,66)(65,82)(69,75)(71,85)(72,83)(73,84)(76,80)(78,81);(54,55)(56,66,70,61)(57,63,58,72)(59,65)(60,64)(62,71,80,73)(67,83)(68,77)(69,82,74,78)(75,81)(76,84)(79,85)

Polycyclic presentation:

2^8,53|a^2=g,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=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, 178780);