Finite Group Databasebeta

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#72275087

order2304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Splittings

FittingC3×#3025 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#10593630
commutator, cosocle, hypocenterC3×#997C23
FrattiniC23C22×C3⋊S4
FittingC3×#3025D3
socleC2×C3C2×#1020⋊C2
hypercenterC4#10593630
LCSG ▷ C3×#997
UCSC1 ◁ C2 ◁ C4

1346 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 1346 conjugacy classes of subgroups — too many to draw. Only its 57 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)(4,13,7,6)(5,18,11,16)(8,14,15,12)(9,19,17,10);(6,15,13,12,7,14)(10,17,19,16,11,18);(1,2,3)(4,5,8,9)(6,10,12,16)(7,11,15,17)(13,19,14,18)

Polycyclic presentation:

2^3,3^2,2^5|a^2=j,f^2=j,g^2=j,c^a=cj,d^c=d^2,e^c=e^2,f^b=fij,f^c=fghij,f^e=g,g^b=ghj,g^c=gh,g^e=fgj,g^f=gj,h^e=hi,h^f=hj,i^c=hi,i^e=h,i^g=ij

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#11714175, #72275094, #72275114
maximal normal subgroup ofD3×#11714175, #72275094, #72275114

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