Finite Group Databasebeta

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#72242418

order3456
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

FittingC3×H(5,𝔽2) ⋊ D32
derivedC3×Ω+(4,𝔽3) ⋊ C22

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#13181790
commutator, cosocle, hypocenterC3×Ω+(4,𝔽3)C22
FittingC3×H(5,𝔽2)D32
socleC2×C3S42
LCSG ▷ C3×Ω+(4,𝔽3)
UCSC1 ◁ C2

706 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

(1,31,15,5)(2,28,4,14)(3,16,25,12)(6,27,11,18)(7,29,17,21)(8,32,24,22)(9,19,13,26)(10,30,23,20)(33,34,35);(2,7)(3,9)(4,21,31,15,20,32)(5,30,25,18,19,24)(8,10)(11,26,16,17,14,27)(12,29,23)(13,22,28)(34,35)

Polycyclic presentation:

2^2,3^3,2^5|f^2=j,g^2=j,c^a=c^2,c^b=c^2,d^a=c^2d,d^b=d^2,e^a=e^2,f^a=fj,f^b=fj,f^c=gj,f^d=fgj,g^a=fgj,g^b=fgj,g^c=fg,g^d=fj,g^f=gj,h^a=fhj,h^b=fh,h^c=fh,h^d=i,h^f=hj,i^a=ij,i^b=ghij,i^c=fgij,i^d=hi,i^g=ij

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#11923456, #72242441, #72252930
maximal normal subgroup ofD3×#11923456, #72242441, #72252930

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