Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C72×C8 ⋊ C2×C3 |
| Hall | C7×#6467 ⋊ C3 |
| Hall | #3861 ⋊ #135 |
| Sylow | C72 ⋊ C3×#135 |
314 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,3,10,16,5,12,4,11)(2,7,13,15,6,14,8,9)(18,19,20)(21,23,22)(24,25,26)(27,28,30);(2,6)(7,14)(8,13)(9,15)(17,18,20,22,19,21,23);(1,2)(3,9)(4,13)(5,6)(7,11)(8,10)(12,15)(14,16)(25,27)(26,29)(28,30)Polycyclic presentation:
2^3,3,2^2,7^2|c^2=ef,e^2=f,b^a=bf,c^a=ce,e^a=ef,g^a=g^6,g^d=g^4,h^a=gh,h^d=h^4182 conjugacy classes — view the full character table.
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, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);