Finite Group DatabasebetaFinite Group Databasebeta| order | 6912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D3×#11923456 |
| Fitting | C3×H(5,𝔽2) ⋊ C2×D32 |
| derived | C3×Ω+(4,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×S42 |
| commutator, cosocle, hypocenter | C3×Ω+(4,𝔽3) | C23 |
| Fitting | C3×H(5,𝔽2) | C2×D32 |
| socle | C2×C3 | C2×S42 |
| LCS | G ▷ C3×Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2 | |
3315 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 3315 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,31,15,5)(2,28,4,14)(3,16,25,12)(6,27,11,18)(7,29,17,21)(8,32,24,22)(9,19,13,26)(10,30,23,20)(33,34,35);(4,20,31)(5,19,25)(11,14,16)(12,29,23)(13,22,28)(15,21,32)(17,26,27)(18,30,24)(34,35);(2,7)(3,9)(4,15)(5,18)(8,10)(11,17)(14,26)(16,27)(19,30)(20,21)(24,25)(31,32)102 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,31,15,5)(2,28,4,14)(3,16,25,12)(6,27,11,18)(7,29,17,21)(8,32,24,22)(9,19,13,26)(10,30,23,20)(33,34,35), (4,20,31)(5,19,25)(11,14,16)(12,29,23)(13,22,28)(15,21,32)(17,26,27)(18,30,24)(34,35), (2,7)(3,9)(4,15)(5,18)(8,10)(11,17)(14,26)(16,27)(19,30)(20,21)(24,25)(31,32));