Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
| direct product | C3×ΓL(1,𝔽8)×(C3⋊D3)⋊C3 |
| Fitting | C3×C7×H(3,𝔽3) ⋊ C2×C3 |
| derived | C32×C7 ⋊ C2×C33 |
| Hall | C3×ΓL(1,𝔽8)×H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ C32×(C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8)×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C32×C7 | C2×C33 |
| Frattini | C3 | C32×D3×ΓL(1,𝔽8) |
| Fitting | C3×C7×H(3,𝔽3) | C2×C3 |
| socle | C32×C7 | C32×D3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C3 | |
720 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,5,4,6,7)(12,19,15,14,16,18)(13,17);(8,9,10)(11,12,19)(13,16,14)(15,18,17);(2,3,4)(5,7,6)150 conjugacy classes — view the full character table.
| maximal subgroup of | C3×ΓL(1,𝔽8)×#4183, ΓL(1,𝔽8)×#4207⋊C3, ΓL(1,𝔽8)×#92300, … |
| maximal normal subgroup of | C3×AGL(1,𝔽7)×(C3⋊D3)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,5,4,6,7)(12,19,15,14,16,18)(13,17), (8,9,10)(11,12,19)(13,16,14)(15,18,17), (2,3,4)(5,7,6));