Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#72227458 |
| Fitting, Sylow | C2×#346282 ⋊ C3 |
| derived | #346282 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72227458 |
| commutator, cosocle, hypocenter | #346282 | C2×C3 |
| Frattini | C25 | C2×C22⋊A4 |
| Fitting | C2×#346282 | C3 |
| socle | C23 | #73293 |
| LCS | G ▷ #346282 | |
| UCS | C1 ◁ C2 | |
2192 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 2192 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) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,7,6,11,22)(3,9,28,14,31,45)(4,34,23)(5,10,38)(8,41,32)(12,26,24)(13,30,46)(15,16,40)(17,36,44,21,35,42)(18,19,25)(20,29,47)(27,43,33,39,48,37);(1,5,6,18)(2,11)(3,20,14,13)(4,17,15,21)(7,38,22,25)(8,26)(10,19)(12,33,32,37)(16,34)(23,44,40,42)(24,48,41,43)(27,39)(28,46,45,47)(35,36);(1,12)(2,16)(3,4)(5,37,18,33)(6,32)(7,28)(8,9)(10,35,19,36)(11,34)(13,21,20,17)(14,15)(22,45)(23,24)(25,47,38,46)(26,31)(27,30,39,29)(40,41)(42,43,44,48)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((1,2,7,6,11,22)(3,9,28,14,31,45)(4,34,23)(5,10,38)(8,41,32)(12,26,24)(13,30,46)(15,16,40)(17,36,44,21,35,42)(18,19,25)(20,29,47)(27,43,33,39,48,37), (1,5,6,18)(2,11)(3,20,14,13)(4,17,15,21)(7,38,22,25)(8,26)(10,19)(12,33,32,37)(16,34)(23,44,40,42)(24,48,41,43)(27,39)(28,46,45,47)(35,36), (1,12)(2,16)(3,4)(5,37,18,33)(6,32)(7,28)(8,9)(10,35,19,36)(11,34)(13,21,20,17)(14,15)(22,45)(23,24)(25,47,38,46)(26,31)(27,30,39,29)(40,41)(42,43,44,48));