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#72274985

order2304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC3×#585 ⋊ C2×D3

Subgroups

subgroupG/H
centerC2C2×#10593244
commutator, cosocle, hypocenterC3×#997C23
FrattiniC23C2×D3×S4
FittingC3×#585C2×D3
socleC2×C3C2×#1020⋊C2
hypercenterC4#10593244
LCSG ▷ C3×#997
UCSC1 ◁ C2 ◁ C4

1219 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1219 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

(2,3)(4,12,14,7)(5,18,16,17)(6,15,8,13)(9,19,10,11);(1,2,3)(5,9)(6,12)(7,15)(10,18)(13,14)(16,19);(4,5,8,9)(6,10,14,16)(7,11,15,17)(12,19,13,18)

Polycyclic presentation:

2^3,3^2,2^5|a^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,d^c=d^2,e^b=e^2,e^c=e^2,f^b=fg,f^c=fghj,f^e=g,g^b=gj,g^c=ghi,g^e=fgj,g^f=gj,h^b=i,h^c=i,h^e=i,h^f=hj,h^g=hj,i^b=h,i^c=h,i^e=hi,i^g=ij

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72274997, #72275020, D3×#11714175
maximal normal subgroup of#72274997, #72275020, D3×#11714175

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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);