Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
| direct product | ΓL(1,𝔽8)×C3⋊#272 |
| Fitting, derived | C3×C7×H(3,𝔽3) ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8)×H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ C3×C3⋊#272 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | ΓL(1,𝔽8)×C3⋊(C3⋊D3) |
| commutator, Fitting, cosocle, hypocenter | C3×C7×H(3,𝔽3) | C2×C3 |
| socle | C32×C7 | C3×C3⋊D3 |
| LCS | G ▷ C3×C7×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
834 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 834 conjugacy classes of subgroups — too many to draw. Only its 97 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 — 4 generators on 19 points:
(2,3)(5,7)(8,12)(9,11)(14,15,16)(17,19,18);(1,2,3)(13,14,15,17,16,18,19);(5,8,9)(7,11,12);(4,5,7)(6,9,11)(8,12,10)105 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#92300, AGL(1,𝔽7)×C3⋊#272 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#92300, AGL(1,𝔽7)×C3⋊#272 |
| maximal quotient of | ΓL(1,𝔽8)×#92338 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(5,7)(8,12)(9,11)(14,15,16)(17,19,18), (1,2,3)(13,14,15,17,16,18,19), (5,8,9)(7,11,12), (4,5,7)(6,9,11)(8,12,10));