#27536903

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2126 ⋊ C₂
Sylow	C₅₃ ⋊ #33361

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₅₃×#2126	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#20288502	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	#2126	D₅₃
hypocenter	C₅₃	#33361
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #2126

1252 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)(57,68)(61,75)(65,79)(69,81)(73,84)(76,82)(78,85)(80,83);(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)(54,57)(55,61)(56,65)(58,69)(59,68)(60,73)(62,76)(63,75)(64,78)(66,80)(67,79)(70,82)(71,83)(72,81)(74,84)(77,85);(54,56,59,67)(55,60,63,74)(57,65,68,79)(58,71,72,66)(61,73,75,84)(62,77,70,64)(69,83,81,80)(76,85,82,78);(54,55)(56,64,67,77)(57,61)(58,70,72,62)(59,63)(60,66)(65,78,79,85)(68,75)(69,82,81,76)(71,74)(73,80)(83,84)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322379);