#27498613

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1912 ⋊ C₂
Sylow	C₅₃ ⋊ #30442

Subgroups

	subgroup	G/H
center	C₂	#20280921
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1912	C₂
socle	C₂×C₅₃	#2668
prime radical	#20284088	C₂
hypercenter	#1912	D₅₃
hypocenter	C₅₃	#30442
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1912

878 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)(54,57)(55,63)(56,68)(58,71)(59,70)(60,72)(61,75)(62,76)(64,79)(65,78)(66,80)(67,81)(69,84)(73,83)(74,82)(77,85);(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)(55,62)(57,68)(58,67)(59,73)(60,69)(61,74)(63,76)(64,66)(65,77)(70,83)(71,81)(72,84)(75,82)(78,85)(79,80);(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)(55,61)(56,67)(57,70)(58,59)(63,78)(64,65)(66,74)(69,73)(71,72)(76,82)(80,85)(81,83);(54,55,60,65)(56,66,69,74)(57,63,72,78)(58,61,59,64)(62,73,77,67)(68,82,84,80)(70,79,71,75)(76,81,85,83)

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 284089);