Finite Group DatabasebetaFinite Group Databasebeta| order | 2646 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C7×#4430 ⋊ C2 |
| Hall | C7⋊D7 ⋊ 3−1+2 |
| Sylow | C72 ⋊ C2×3−1+2 |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,5,9)(4,7,8)(10,11)(12,13)(15,16)(18,19)(20,21)(22,23);(1,5,7,3,9,4,6,2,8)(11,14,12)(13,15,16)(17,23,18,21,20,19,22)Polycyclic presentation:
2,3^3,7^2|b^3=d,c^b=cd^2,e^a=e^6,e^b=e^5f^3,f^a=f^6,f^b=e118 conjugacy classes — view the full character table.
| maximal subgroup of | #72224657, D7×#4430⋊C2, C2×#72224638 |
| maximal normal subgroup of | #72224657, D7×#4430⋊C2, C2×#72224638 |
| maximal quotient of | C2×#72224638 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 7, 7]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 2, F.5^5*F.6^3);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);