Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C23×C3×D4 ⋊ C2×D3 |
3534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,4,2,3)(5,6);(2,3,4)(9,13)(12,15);(9,12)(10,14)(13,15);(6,7)(8,9,10,12,11,13,14,15)Polycyclic presentation:
2^6,3^2,2^2|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^2,g^d=g^2,h^a=g^2h,h^d=h^2,i^h=j,j^d=ij,j^h=ij87 conjugacy classes — view the full character table.
| maximal subgroup of | #72258669, D3×S4×AGL(1,ℤ/8) |
| maximal normal subgroup of | D3×S4×AGL(1,ℤ/8) |
| maximal quotient of | #72258669 |
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, 2, 2, 2, 3, 3, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);