Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
| direct product | C32×C3⋊D3×ΓL(1,𝔽8) |
| Fitting, socle | C34×C7 ⋊ C2×C3 |
| derived | C32×C7 ⋊ C2×C33 |
| Hall | C34×ΓL(1,𝔽8) ⋊ C2 |
| Sylow | C7 ⋊ C33×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊D3×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C32×C7 | C2×C33 |
| Frattini | C1 | G |
| Fitting, socle | C34×C7 | C2×C3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C32 | |
3168 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 3168 conjugacy classes of subgroups — too many to draw. Only its 238 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)(11,12,13)(15,16)(18,19);(8,9,10)(17,18,19);(2,3,4)(5,7,6)(14,15,16)270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×ΓL(1,𝔽8)×#4183, ΓL(1,𝔽8)×#92300, ΓL(1,𝔽8)×#92302, … |
| maximal normal subgroup of | C32×C3⋊D3×AGL(1,𝔽7) |
| maximal quotient of | ΓL(1,𝔽8)×#4207⋊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)(11,12,13)(15,16)(18,19), (8,9,10)(17,18,19), (2,3,4)(5,7,6)(14,15,16));