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#72228152

order3888
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C3

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C3⋊#10620185
commutator, prime radical, cosocle, hypocenterC3×SL(2,ℤ/9)C2
Fitting, socleC2×C34S4
LCSG ▷ C3×SL(2,ℤ/9)
UCSC1 ◁ C2

242 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 15 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) = 28
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 28 points:

(2,3)(4,6)(5,12)(7,8,10,11)(13,28,16,21,17,18,26,27)(14,15,19,22,20,23,24,25);(1,2,3);(4,5,7)(6,8,10)(9,11,12)(15,22,16)(18,27,19)(21,24,28)(23,25,26)

Polycyclic presentation:

2,3^2,2^3,3^3|a^2=f,d^2=f,e^2=f,b^a=b^2,c^a=c^2,d^a=de,d^b=e,e^a=ef,e^b=def,e^d=ef,g^a=i,g^b=i,g^d=g^2,g^e=g^2,h^a=ghi^2,h^b=g^2i^2,h^d=h^2,h^e=g^2h,i^a=g,i^b=g^2h^2,i^e=i^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228153

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");;
gap> coll := SingleCollector(F, [2, 3, 3, 2, 2, 2, 3, 3, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.4*F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.9);;
gap> SetConjugate(coll, 7, 2, F.9);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.9^2);;
gap> SetConjugate(coll, 8, 2, F.7^2*F.9^2);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.7^2*F.8);;
gap> SetConjugate(coll, 9, 1, F.7);;
gap> SetConjugate(coll, 9, 2, F.7^2*F.8^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);