Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C5 |
No aliases for this group.
| Hall | C3×C52 ⋊ C2×M16 |
| Sylow | C52 ⋊ C2×ΓL(1,𝔽25) |
| Sylow | C3 ⋊ #11717448 |
380 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(4,6,8,7)(5,10,9,11)(12,13,16,14,15)(17,18,19,21,20);(2,3)(5,9)(10,11);(4,5,7,11,8,9,6,10)(13,14,15,16)(18,19,20,21)Polycyclic presentation:
2^5,3,5^2|a^2=d,c^2=e,d^2=e,b^a=be,f^b=f^2,g^a=g^2,g^d=g^4,h^a=g^3h^3,h^d=h^4108 conjugacy classes — view the full character table.
| maximal subgroup of | #72230504, #72230521, D3×#11717448, … |
| maximal normal subgroup of | #72230504, #72230521, D3×#11717448, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^3*F.8^3);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);