#27512548

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2080 ⋊ C₂
Sylow	C₅₃ ⋊ #31420

Subgroups

	subgroup	G/H
center	C₂	#20281144
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2080	C₂
socle	C₂×C₅₃	#2688
prime radical	#20284446	C₂
hypercenter	#2080	D₅₃
hypocenter	C₅₃	#31420
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2080

670 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)(55,62)(56,67)(58,74)(60,76)(61,69)(63,64)(65,77)(66,82)(70,73)(75,85)(78,81)(79,80);(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)(55,61,64,79)(56,66)(58,73,74,70)(60,75)(62,69,63,80)(65,78)(67,82)(68,84)(72,83)(76,85)(77,81);(54,56,58,72,59,67,74,68)(55,60,63,76,64,65,62,77)(57,66,70,84,71,82,73,83)(61,78,69,85,79,75,80,81);(54,55)(56,65)(57,69)(58,62)(59,64)(60,67)(61,70)(63,74)(66,81)(68,77)(71,80)(72,76)(73,79)(75,84)(78,83)(82,85)

Polycyclic presentation:

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

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 298024);