Finite Group DatabasebetaFinite Group Databasebeta| order | 2058 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C73 ⋊ C2×C3 |
| Hall | C7⋊#3862 ⋊ C2 |
| Hall | C72×D7 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C73 | C2×C3 |
| LCS | G ▷ C73 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 294 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,3,4)(5,7,6)(9,10,11)(12,14,13)(16,20,19,17,18,21);(8,9,10,12,11,13,14)(15,16,19,20,18,21,17);(1,2,3,5,4,6,7)Polycyclic presentation:
2,3,7^3|c^b=c^2,d^b=d^2,e^a=e^6,e^b=e^487 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3861, ΓL(1,𝔽8)×#64422, #72241659, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#3861, ΓL(1,𝔽8)×#64422, #72241659, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 3, 7, 7, 7]);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 2, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);