Finite Group DatabasebetaFinite Group Databasebeta| order | 7938 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Hall | #72224635 ⋊ C2 |
| Hall | C7⋊D7 ⋊ C3×3−1+2 |
| Sylow | C72 ⋊ C2×C3×3−1+2 |
416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 3 generators on 23 points:
(10,13,15,11,12,16)(18,22,21,19,20,23);(2,5,9)(4,7,8)(17,18,21,22,20,23,19);(1,5,7,3,9,4,6,2,8)(11,14,12)(13,15,16)Polycyclic presentation:
2,3^4,7^2|b^3=e,c^b=ce^2,f^a=f^6,f^b=f^5g^3,f^d=f^4,g^a=g^6,g^b=f,g^d=g^4106 conjugacy classes — view the full character table.
| maximal subgroup of | #72229823, C2×#72224657 |
| maximal normal subgroup of | #72229823, C2×#72224657 |
| maximal quotient of | C2×#72224657 |
| holomorph of | C7⋊C9 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 7, 7]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.6^5*F.7^3);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);