Finite Group DatabasebetaFinite Group Databasebeta| order | 46656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D3×#72210382 |
| Fitting, socle | C35 ⋊ C22×GL(2,𝔽3) |
| derived | C3×#14519481 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#14519481 | C23 |
| Fitting, socle | C35 | C22×GL(2,𝔽3) |
| LCS | G ▷ C3×#14519481 | |
| UCS | C1 | |
5467 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 5467 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(4,6,12,8,9,11,10,7)(13,14,15,18,21,19,17,20);(2,3)(5,6,11)(7,9,10)(14,18,21)(15,16,17);(14,17)(15,18)(16,21)(19,20)99 conjugacy classes — view the full character table.
| holomorph of | C3×C3⋊D3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,6,12,8,9,11,10,7)(13,14,15,18,21,19,17,20), (2,3)(5,6,11)(7,9,10)(14,18,21)(15,16,17), (14,17)(15,18)(16,21)(19,20));