Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C52 ⋊ #103 |
| Sylow | C52 ⋊ #816 |
| Sylow | C3 ⋊ #11716767 |
220 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 2 generators on 21 points:
(2,3)(5,9)(6,7)(10,11)(12,13,15,14)(17,18,21,20);(1,2,3)(4,5,6,9,8,11,7,10)(13,14,15,16)(18,19,21,20)Polycyclic presentation:
2^5,3,5^2|a^2=c,b^2=cd,c^2=e,d^2=e,b^a=bd,c^b=ce,d^b=de,f^b=f^2,g^a=g^2,g^b=g^3,g^c=g^4,h^a=g^3h^3,h^b=h^3,h^c=h^472 conjugacy classes — view the full character table.
| maximal subgroup of | #72230760, #72230800, D3×#11716767, … |
| maximal normal subgroup of | #72230760, #72230800, D3×#11716767, … |
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.3);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^3*F.8^3);;
gap> SetConjugate(coll, 8, 2, F.8^3);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);