#27472480

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3058 ⋊ C₂
Sylow	C₅₃ ⋊ #28183

Subgroups

	subgroup	G/H
center	C₂	#20279614
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3058	C₂
socle	C₂×C₅₃	#2535
hypercenter	#3058	D₅₃
hypocenter	C₅₃	#28183
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3058

812 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)(54,57,59,71,60,72,58,70)(55,63,65,79,64,78,61,74)(56,67,73,83,69,82,68,81)(62,76,75,84,77,85,66,80);(54,56)(55,62)(57,67)(58,68)(59,73)(60,69)(61,66)(63,76)(64,77)(65,75)(70,81)(71,83)(72,82)(74,80)(78,85)(79,84);(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)(54,55,59,65,60,64,58,61)(56,66,68,77,69,75,73,62)(57,63,71,79,72,78,70,74)(67,80,81,85,82,84,83,76);(55,61)(57,70)(58,59)(62,75)(63,78)(64,65)(66,77)(67,81)(68,73)(71,72)(80,84)(82,83)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 257956);