#27415625

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2017 ⋊ C₂
Sylow	C₅₃ ⋊ #23894

Subgroups

	subgroup	G/H
center	C₂	#20278024
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2017	C₂
socle	C₂×C₅₃	#2403
hypercenter	#2017	D₅₃
hypocenter	C₅₃	#23894
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2017

1642 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)(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);(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)(55,62)(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);(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);(54,55,60,65)(56,66,71,79)(57,72,75,82)(58,76,59,64)(61,70,80,69)(62,73,63,74)(67,83,84,77)(68,78,85,81)

Polycyclic presentation:

2^8,53|a^2=h,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,f^c=fh,g^c=gh,e^d=eh,i^d=i^52

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 201101);