Finite Group Databasebeta

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#72275061

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2D3×#1020⋊C2
commutator, cosocle, hypocenterC3×#997C23
FrattiniC23C2×D3×S4
FittingC3×B(4,𝔽2)C2×D3
socleC2×C3C2×#1020⋊C2
LCSG ▷ C3×#997
UCSC1 ◁ C2

1224 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 40 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)(4,14,6,7)(5,19,10,18)(8,15,13,16)(9,12,17,11);(1,2,3)(6,13)(7,16)(10,17)(11,18)(12,19)(14,15);(4,5)(6,12)(7,11)(8,9)(10,14)(13,19)(15,17)(16,18)

Polycyclic presentation:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72275070, #72275082, D3×#11714175
maximal normal subgroup of#72275070, #72275082, 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, 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, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8*F.9*F.10);;
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, 1, F.7*F.8);;
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, 1, F.8*F.9);;
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);