Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, derived | C3×C7×H(3,𝔽3) ⋊ C2×C3 |
| Hall | (C3×ΓL(1,𝔽8))⋊H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ #92300 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊((C3⋊D3)⋊ΓL(1,𝔽8)) |
| commutator, Fitting, cosocle, hypocenter | C3×C7×H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3⋊D3×ΓL(1,𝔽8) |
| socle | C32×C7 | C3×C3⋊D3 |
| LCS | G ▷ C3×C7×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
384 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 36 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:
2,3^4,7,3|c^a=c^2,c^b=ce,d^a=d^2,d^c=dg,e^a=e^2,f^b=f^281 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#92300, #72269700 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#92300, #72269700 |
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, 3]);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);