#27411016

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2004 ⋊ C₂
Sylow	C₅₃ ⋊ #23401

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2004	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288046	C₂
hypercenter	#2004	D₅₃
hypocenter	C₅₃	#23401
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2004

1954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 89 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)(61,72)(65,81)(69,86)(73,76)(78,80)(82,84)(85,89)(87,88);(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)(55,57)(58,61)(59,65)(60,69)(62,76)(63,72)(64,78)(66,84)(67,81)(68,85)(70,88)(71,86)(73,77)(74,82)(75,87)(79,80)(83,89);(58,60,63,71)(59,64,67,79)(61,69,72,86)(62,75,77,70)(65,80,81,78)(66,83,74,68)(73,88,76,87)(82,89,84,85);(54,55)(56,57)(58,59)(60,68)(61,65)(62,74)(63,67)(64,70)(66,77)(69,85)(71,83)(72,81)(73,84)(75,79)(76,82)(78,88)(80,87)(86,89)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 196492);