Finite Group DatabasebetaFinite Group Databasebeta| order | 6174 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C73 ⋊ C2×C32 |
| Hall | #72241579 ⋊ C2 |
| Hall | C7×C7⋊D7 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C73 | C2×C32 |
| LCS | G ▷ C73 | |
| UCS | C1 | |
124 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2,3^2,7^3|d^a=d^6,d^b=e,d^c=d^2,e^a=e^6,e^b=d^5e^3,e^c=e^2,f^b=f^254 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72241714 |
| maximal normal subgroup of | ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72241714 |
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, 7, 7, 7]);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 5, 2, F.4^5*F.5^3);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);