#27415645

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2022 ⋊ C₂
Sylow	C₅₃ ⋊ #23896

Subgroups

	subgroup	G/H
center	C₂	#20278024
commutator, cosocle	C₂⁴×C₅₃	C₂⁴
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#2022	C₂
socle	C₂×C₅₃	#2403
hypercenter	#2022	D₅₃
hypocenter	C₅₃	#23896
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #2022

1764 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)(54,58)(55,62)(56,65)(57,67)(59,69)(60,71)(61,74)(63,76)(64,77)(66,79)(68,81)(70,82)(72,83)(73,84)(75,80)(78,85);(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)(54,57)(55,61)(56,64)(58,67)(59,68)(60,70)(62,74)(63,75)(65,77)(66,78)(69,81)(71,82)(72,73)(76,80)(79,85)(83,84);(54,56,59,66)(55,60,63,72)(57,64,68,78)(58,65,69,79)(61,73,75,70)(62,71,76,83)(67,77,81,85)(74,84,80,82);(54,55)(56,60)(57,61)(58,62)(59,63)(64,70)(65,71)(66,72)(67,80)(68,75)(69,76)(73,78)(74,81)(77,84)(79,83)(82,85)

Polycyclic presentation:

2^8,53|b^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,e^c=eh,f^d=fh,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, 201121);