Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| derived | #2273⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11714268 |
| commutator, hypocenter | #2273⋊C3 | C4 |
| Frattini | C24 | #1020 |
| Fitting | #35390 | D3 |
| socle | C23 | #5936 |
| prime radical, cosocle | #35390⋊C3 | C2 |
| LCS | G ▷ #2273⋊C3 | |
| UCS | C1 ◁ C2 | |
837 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 837 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,4)(2,22,9,17)(5,15)(6,12)(7,16,20,24)(8,18)(10,13)(14,19)(21,23);(1,2,6)(3,12,23)(4,14,24)(5,9,18)(7,17,10)(8,19,11)(13,20,22)(15,21,16)Polycyclic presentation:
2^2,3,2^7|a^2=b,d^2=i,e^2=ij,c^a=c^2,d^a=de,d^b=dhi,d^c=ehi,e^a=ehj,e^b=ehij,e^c=deh,e^d=eh,f^a=ghi,f^c=g,f^d=fh,f^e=fh,g^a=fj,g^c=fg,g^e=gh,i^a=hj,i^c=ij,j^a=hi,j^c=hi34 conjugacy classes — view the full character table.
| maximal subgroup of | #72247525 |
| maximal quotient of | #72266406 |
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, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);