Finite Group Databasebeta

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#72242413

order3456
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D3×A4×S4
commutator, cosocle, hypocenterC3×#994C22×C3
FittingC3×H(5,𝔽2)C2×C3×D3
socleC2×C3C2×A4×S4
LCSG ▷ C3×#994
UCSC1 ◁ C2

834 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 51
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 51 points:

(1,4,7,8,20,5)(2,40,34)(3,16,14,17,38,21)(6,26,25,27,23,41)(9,45,32)(10,31,44)(11,47,35)(12,24,48)(13,46,29)(15,39)(18,22)(19,37)(28,43,33)(30,42,36)(50,51);(1,31,8,13)(2,18,9,22)(3,35,17,36)(4,28,20,12)(5,11,7,30)(6,46,27,44)(10,16,29,38)(14,48,21,33)(15,32,39,34)(19,47,37,42)(23,40,26,45)(24,25,43,41)(49,50,51)

Polycyclic presentation:

2^2,3,2^2,3^2,2^3|d^2=j,e^2=j,h^2=j,i^2=j,b^a=bj,d^c=e,e^c=dej,e^d=ej,f^a=f^2,g^b=g^2,h^a=hj,h^f=i,i^a=hi,i^f=hij,i^h=ij

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#983⋊A4, #72242440, #72242447
maximal normal subgroup ofD3×#983⋊A4, #72242440, #72242447

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