#27510484

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2053 ⋊ C₂
Sylow	C₅₃ ⋊ #31280

Subgroups

	subgroup	G/H
center	C₄	#16811365
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2053	C₂
socle	C₂×C₅₃	#2684
prime radical	#20284061	C₂
hypercenter	#2053	D₅₃
hypocenter	C₅₃	#31280
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2053

740 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)(54,55)(56,68)(57,62)(58,64)(59,63)(60,65)(61,70)(66,71)(67,83)(69,78)(72,75)(73,77)(74,81)(76,80)(79,85)(82,84);(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)(55,62)(56,67)(58,74)(59,75)(60,76)(61,77)(63,72)(64,81)(65,80)(66,79)(68,83)(69,84)(70,73)(71,85)(78,82);(54,56,59,70,60,71,58,69)(55,61,64,68,65,78,63,66)(57,73,74,67,76,84,75,85)(62,79,72,82,80,83,81,77);(54,55)(56,66)(57,72)(58,64)(59,63)(60,65)(61,69)(62,74)(67,82)(68,71)(70,78)(73,83)(75,80)(76,81)(77,85)(79,84)

Polycyclic presentation:

2^8,53|b^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,f^c=fh,g^c=gh,e^d=eh,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, 295960);