#27538488

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2212 ⋊ C₂
Sylow	C₅₃ ⋊ #33595

Subgroups

	subgroup	G/H
center	C₂	#459×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2212	C₂
socle	C₂×C₅₃	C₂×#459
hypercenter	#2212	D₅₃
hypocenter	C₅₃	#33595
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2212

706 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)(55,62)(60,73)(61,76)(63,79)(72,78)(74,84)(75,80)(77,85);(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)(54,56)(55,61)(57,65)(58,70)(59,67)(60,72)(62,76)(63,80)(64,69)(66,68)(71,81)(73,78)(74,77)(75,79)(82,83)(84,85);(54,55,57,60,59,62,68,73)(56,61,65,72,67,76,66,78)(58,63,69,74,71,79,83,84)(64,77,81,75,82,85,70,80);(55,60)(56,64)(57,68)(58,69)(61,75)(62,73)(63,79)(65,70)(66,81)(67,82)(71,83)(72,77)(76,80)(78,85)

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323964);