Finite Group Databasebeta

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#72233239

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

Aliases

No aliases for this group.

Splittings

FittingC3×B(4,𝔽2) ⋊ C2×D3

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

1304 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 1304 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:

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

Polycyclic presentation:

2^3,3^2,2^5|b^a=bj,c^b=cj,d^c=d^2,e^a=e^2,f^a=g,f^b=fhij,f^e=g,g^a=f,g^b=ghj,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#11714170, #72275031, #72275082
maximal normal subgroup ofD3×#11714170, #72275031, #72275082

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