#27536850

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#352 ⋊ C₂
Sylow	C₅₃ ⋊ #33350

Subgroups

	subgroup	G/H
center	C₂	C₂×#99×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#352	C₂
socle	C₂×C₅₃	C₂²×#99
prime radical	#20288353	C₂
cosocle	C₂×D₄×C₅₃	C₂⁴
hypercenter	C₂×#352	D₅₃
hypocenter	C₅₃	#33350
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#352

1240 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:

(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)(57,68)(61,74)(65,78)(69,81)(72,83)(75,82)(76,84)(79,85);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57)(55,61)(56,65)(58,69)(59,68)(60,72)(62,75)(63,74)(64,76)(66,79)(67,78)(70,82)(71,81)(73,83)(77,84)(80,85);(54,56,59,67)(55,60,63,73)(57,65,68,78)(58,66,71,80)(61,72,74,83)(62,64,70,77)(69,79,81,85)(75,76,82,84);(54,55)(56,64)(57,61)(58,70,71,62)(59,63)(60,66,73,80)(65,76)(67,77)(68,74)(69,82,81,75)(72,79,83,85)(78,84)

Polycyclic presentation:

2^8,53|a^2=f,b^2=h,b^a=be,e^a=eg,g^a=gh,f^b=fg,d^c=dh,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, 322326);