Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽8)⋊#272 |
| Fitting | C3×C7×H(3,𝔽3) ⋊ C2×C3 |
| derived | C7×H(3,𝔽3) ⋊ C2×C32 |
| Hall | C3×ΓL(1,𝔽8)×H(3,𝔽3) ⋊ C2 |
| Sylow | C7 ⋊ C32×#272 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | C3⋊(C3⋊AGL(1,𝔽7)) |
| commutator, cosocle, hypocenter | C7×H(3,𝔽3) | C2×C32 |
| Frattini | C3 | C3×C3⋊(C3⋊AGL(1,𝔽7)) |
| Fitting | C3×C7×H(3,𝔽3) | C2×C3 |
| socle | C32×C7 | C3×C3⋊D3 |
| LCS | G ▷ C7×H(3,𝔽3) | |
| UCS | C1 ◁ C32 | |
888 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 189 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,7)(8,12)(9,11)(13,14)(15,16)(17,19);(5,8,9)(7,11,12)(14,16,18)(15,17,19);(4,5,7)(6,9,11)(8,12,10)123 conjugacy classes — view the full character table.
| maximal subgroup of | #72254918, C3×AGL(1,𝔽7)×#272 |
| maximal normal subgroup of | C3×AGL(1,𝔽7)×#272 |
| maximal quotient of | C3×#72249208, #72254872, #72254880, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,7)(8,12)(9,11)(13,14)(15,16)(17,19), (5,8,9)(7,11,12)(14,16,18)(15,17,19), (4,5,7)(6,9,11)(8,12,10));