Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #13096134 ⋊ C3 |
| Hall | #3861 ⋊ #106 |
| Sylow | C72 ⋊ C3×#106 |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1568 |
A minimal faithful permutation representation — 3 generators on 26 points:
(5,6,7,9,8,10,11,12)(14,16,17)(15,18,19)(20,21,26)(22,23,25);(1,2,3,4)(6,9)(7,11)(10,12)(14,15)(16,18)(17,19)(21,22)(23,25)(24,26);(13,14,17,18,16,19,15)Polycyclic presentation:
2^2,3,2^3,7^2|a^2=d,b^2=e,e^2=f,b^a=be,e^a=ef,g^a=g^6,g^c=g^4,h^a=h^6,h^c=h^4170 conjugacy classes — view the full character table.
| maximal subgroup of | #106×#64424, #72259955, #72260015 |
| maximal normal subgroup of | #106×#64424, #72259955, #72260015 |
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, 3, 2, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);