#27431983

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2038 ⋊ C₂
Sylow	C₅₃ ⋊ #25183

Subgroups

	subgroup	G/H
center	C₂²	#16811255
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2038	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133263	C₂²
hypercenter	#2038	D₅₃
hypocenter	C₅₃	#25183
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2038

1000 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)(58,62,64,61)(59,67,70,86)(60,73,75,76)(63,77,81,78)(65,82,84,89)(66,69,79,68)(71,87,83,85)(72,80,88,74);(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)(54,55)(56,57)(58,61,64,62)(59,66,70,79)(60,72,75,88)(63,77,81,78)(65,82,84,89)(67,69,86,68)(71,85,83,87)(73,80,76,74);(54,55,56,57)(58,60)(59,65)(61,76)(62,73)(63,80)(64,75)(66,85)(67,82)(68,83)(69,71)(70,84)(72,78)(74,81)(77,88)(79,87)(86,89);(54,55,56,57)(58,59)(60,71)(61,66)(62,79)(63,69)(64,70)(65,74)(67,78)(68,81)(72,85)(73,82)(75,83)(76,89)(77,86)(80,84)(87,88)

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 217459);