Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#72227467 |
| Fitting, Sylow | C2×#346284 ⋊ C3 |
| derived | #346284 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72227467 |
| commutator, cosocle, hypocenter | #346284 | C2×C3 |
| Frattini | C25 | C2×C22⋊A4 |
| Fitting | C2×#346284 | C3 |
| socle | C23 | #73293 |
| LCS | G ▷ #346284 | |
| UCS | C1 ◁ C2 | |
3002 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 3002 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2)(3,11,9,8,7,4)(5,14,29,27,24,17)(6,33,18,28,34,15)(10,12)(13,22,30,16,26,32)(19,20,25,21,31,23);(3,17,8,20)(4,29,9,31)(5,26,25,6)(7,30,11,34)(10,33,12,32)(13,21,18,14)(15,22,16,28)(19,27,24,23);(4,8)(5,16)(6,21)(11,12)(13,15)(17,31)(18,23)(22,24)(25,27)(30,33)64 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((1,2)(3,11,9,8,7,4)(5,14,29,27,24,17)(6,33,18,28,34,15)(10,12)(13,22,30,16,26,32)(19,20,25,21,31,23), (3,17,8,20)(4,29,9,31)(5,26,25,6)(7,30,11,34)(10,33,12,32)(13,21,18,14)(15,22,16,28)(19,27,24,23), (4,8)(5,16)(6,21)(11,12)(13,15)(17,31)(18,23)(22,24)(25,27)(30,33));