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#72245704

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
FittingC32×#585D3
socleC2×C3C2×C3×GL(2,ℤ/4)
hypercenter#41((C22⋊C9)⋊C2)⋊C3
hypocenterQ8⋊C9C2×C3×D4
LCSG ▷ #41⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

1261 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1261 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) = 41
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 41 points:

(1,13,20,3,17,29,7,30,18,14,6,27)(2,22,24,8,16,32,4,26,23,11,10,31)(5,28,19,12)(9,25,15,21)(33,41,36,35,34,38,39,37,40);(2,4)(3,11)(5,17)(6,19)(8,14)(9,10)(12,22)(13,21)(15,16)(18,20)(25,30)(26,28)(27,31)(29,32)(34,36)(35,39)(37,40)(38,41);(1,2,7,4)(3,8,14,11)(5,9,19,15)(6,10,17,16)(12,21,28,25)(13,22,30,26)(18,23,20,24)(27,31,29,32)

Polycyclic presentation:

2^3,3,2,3^2,2^3|a^2=j,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#72245730, #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, 1, F.10);;
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);