Finite Group Databasebeta

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#72284927

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#8932767 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72284822
commutator, cosocle#47563⋊C3C22
FrattiniC22×C4C2×#1020
Fitting#8932767D3
socleC23#87524
prime radical#72283832C2
hypocenter#73292D4
LCSG ▷ #47563⋊C3 ▷ #73292
UCSC1 ◁ C2

1667 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 1667 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^3,3,2^7|a^2=ij,b^2=c,c^2=ij,e^2=i,f^2=ik,b^a=bc,c^a=cij,d^b=d^2,e^a=ejk,e^b=fgi,e^c=ej,e^d=fj,f^a=fk,f^b=ehijk,f^c=fjk,f^d=efij,f^e=fij,g^a=gjk,g^b=hj,g^c=gj,g^d=gh,g^e=gjk,g^f=gj,h^a=hj,h^b=gjk,h^c=hk,h^d=g,h^e=hk,h^f=hjk,i^b=ijk,i^d=ik,j^b=k,j^d=jk,k^b=j,k^d=j

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.9*F.10);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.5*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.10);;
gap> SetConjugate(coll, 11, 4, F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);