#27393248

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1782 ⋊ C₂
Sylow	C₅₃ ⋊ #21777

Subgroups

	subgroup	G/H
center	C₂	#20277291
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₅₃×#1782	C₂
socle	C₂×C₅₃	#2308
prime radical	#20288359	C₂
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	#1782	D₅₃
hypocenter	C₅₃	#21777
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #1782

1318 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)(55,62)(56,66)(57,68)(59,71)(60,73)(61,75)(63,70)(64,77)(65,78)(67,80)(69,82)(72,83)(74,84)(76,81)(79,85);(54,57,59,69)(55,61,63,76)(56,65,67,79)(58,68,71,82)(60,72,64,74)(62,75,70,81)(66,78,80,85)(73,83,77,84);(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)(54,56)(55,60)(57,65)(58,66)(59,67)(61,74)(62,73)(63,64)(68,78)(69,79)(70,77)(71,80)(72,76)(75,84)(81,83)(82,85);(54,55)(56,64,67,60)(57,61)(58,70,71,62)(59,63)(65,72,79,74)(66,73)(68,81,82,75)(69,76)(77,80)(78,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 178724);