#27415610

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3019 ⋊ C₂
Sylow	C₅₃ ⋊ #23892

Subgroups

	subgroup	G/H
center	C₂	#20278026
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3019	C₂
socle	C₂×C₅₃	#2403
hypercenter	#3019	D₅₃
hypocenter	C₅₃	#23892
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3019

1674 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 201086);