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#72245749

order3456
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C2×#63694⋊C3
commutator, cosocle#41⋊C9C23×C3
FrattiniC3×C4C22×C3×S4
FittingC2×C32×H(5,𝔽2)D3
socleC2×C3C2×C3×GL(2,ℤ/4)
hypercenterC2×Q8((C22⋊C9)⋊C2)⋊C3
hypocenterQ8⋊C9C2×C3×D4
LCSG ▷ #41⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

1259 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 57 points:

(1,34,36,3,18,39,8,23,41,7,21,19)(2,42,44,10,28,29,9,33,46,14,32,30)(4,16,40,6,22,38,17,20,35,15,5,37)(11,27,43,13,25,45,24,31,48,26,12,47)(49,57,52,51,50,54,55,53,56);(2,9)(3,6)(4,17)(5,19)(7,15)(10,26)(12,29)(13,14)(16,35)(18,37)(20,40)(21,38)(22,39)(23,36)(25,30)(27,43)(28,45)(31,48)(32,47)(33,46)(34,41)(42,44)(50,52)(51,55)(53,56)(54,57);(1,2)(3,10)(4,11)(5,12)(6,13)(7,14)(8,9)(15,26)(16,27)(17,24)(18,28)(19,30)(20,31)(21,32)(22,25)(23,33)(29,39)(34,42)(35,48)(36,44)(37,47)(38,45)(40,43)(41,46)

Polycyclic presentation:

2^3,3,2,3^2,2^3|c^2=j,e^2=j,f^3=g,h^2=j,i^2=j,b^a=bj,c^b=ce,e^b=ej,e^c=ej,f^b=f^2g^2,f^d=fg,g^b=g^2,h^b=ij,h^f=hi,i^b=hj,i^f=h,i^h=ij

Conjugacy classes & character table

127 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245787, #72253040
maximal normal subgroup of#72253040

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 3, 3, 2, 2, 2]);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 6, F.8);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);