Finite Group Databasebeta

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#72275065

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

Aliases

No aliases for this group.

Splittings

FittingC2×C3×B(4,𝔽2) ⋊ C2×C3

Subgroups

subgroupG/H
centerC2C2×D3×#120⋊C3
commutator, cosocle, hypocenterC3×H(5,𝔽2)C23×C3
FrattiniC23C22×D3×A4
FittingC2×C3×B(4,𝔽2)C2×C3
socleC2×C3C22×#120⋊C3
hypercenterC22D3×#120⋊C3
LCSG ▷ C3×H(5,𝔽2)
UCSC1 ◁ C2 ◁ C22

2030 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 2030 conjugacy classes of subgroups — too many to draw. Only its 72 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)(5,9)(6,17,7,13,14,16)(10,18,11)(12,19,15);(1,2,3)(4,17)(5,19)(6,16,13,7)(8,14)(9,18)(10,15,12,11);(4,5)(6,12)(7,15)(8,9)(10,13)(11,16)(14,19)(17,18)

Polycyclic presentation:

2^3,3,2^4,3,2|e^2=j,f^2=j,c^a=cj,e^a=eh,e^b=eh,e^c=eh,e^d=f,f^a=fg,f^b=fg,f^c=fg,f^d=efj,f^e=fj,g^d=ghj,g^e=gj,h^d=g,h^f=hj,i^c=i^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72275070, #72275082, D3×#60399⋊C3
maximal normal subgroup of#72275070, #72275082, D3×#60399⋊C3

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