#27420122

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3057 ⋊ C₂
Sylow	C₅₃ ⋊ #24295

Subgroups

	subgroup	G/H
center	C₂	#20278067
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3057	C₂
socle	C₂×C₅₃	#2408
hypercenter	#3057	D₅₃
hypocenter	C₅₃	#24295
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3057

1302 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)(54,58,59,76,61,77,60,62)(55,64,65,56,67,69,66,70)(57,73,74,82,75,85,78,79)(63,72,71,68,81,83,80,84);(54,57)(55,63)(56,68)(58,73)(59,74)(60,78)(61,75)(62,79)(64,72)(65,71)(66,80)(67,81)(69,83)(70,84)(76,82)(77,85);(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)(54,56,59,69,61,70,60,64)(55,62,65,58,67,76,66,77)(57,72,74,68,75,83,78,84)(63,73,71,82,81,85,80,79);(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)(56,58)(57,71)(59,66)(60,65)(61,67)(62,69)(63,74)(64,76)(68,82)(70,77)(72,85)(73,83)(75,80)(78,81)(79,84)

Polycyclic presentation:

2^8,53|b^2=e,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,i^a=i^52,c^b=cg,g^b=gh,i^b=i^52,f^c=fh,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, 205598);