#27415401

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2003 ⋊ C₂
Sylow	C₅₃ ⋊ #23874

Subgroups

	subgroup	G/H
center	C₂	#20278024
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2003	C₂
socle	C₂×C₅₃	#2403
hypercenter	#2003	D₅₃
hypocenter	C₅₃	#23874
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2003

1674 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)(55,62)(56,65)(58,73)(59,70)(60,68)(61,76)(63,79)(64,78)(66,83)(67,82)(69,71)(72,81)(74,84)(75,80)(77,85);(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)(54,56,59,67)(55,61,64,77)(57,65,70,82)(58,72,71,66)(60,74,63,75)(62,76,78,85)(68,84,79,80)(69,83,73,81);(55,60)(58,71)(61,75)(62,68)(63,64)(66,72)(69,73)(74,77)(76,80)(78,79)(81,83)(84,85);(54,55)(56,61)(57,68)(58,60)(59,64)(62,69)(63,71)(65,80)(66,75)(67,77)(70,79)(72,74)(73,78)(76,81)(82,84)(83,85)

Polycyclic presentation:

2^8,53|c^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,e^d=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, 200877);