Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C8 |
| direct product | C3×#72213975 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #21690 |
| commutator | C4×C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C4×C8 | C24×C3 |
| prime radical | C3×#491500 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C3×#7846 ◁ G | |
2154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2)(3,10)(4,6)(5,8)(7,14)(9,13)(11,15)(12,16)(18,22)(19,24)(20,28)(23,29)(25,30)(26,31)(27,32)(33,34,35);(2,6,8,9)(3,11)(7,15,16,10)(12,14)(18,23,25,26)(19,27)(22,29,30,31)(24,32);(1,3,4,12,5,11,13,14)(2,7,9,15,8,16,6,10);(17,18,24,29,20,26,27,30,21,25,32,31,28,23,19,22)264 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,10)(4,6)(5,8)(7,14)(9,13)(11,15)(12,16)(18,22)(19,24)(20,28)(23,29)(25,30)(26,31)(27,32)(33,34,35), (2,6,8,9)(3,11)(7,15,16,10)(12,14)(18,23,25,26)(19,27)(22,29,30,31)(24,32), (1,3,4,12,5,11,13,14)(2,7,9,15,8,16,6,10), (17,18,24,29,20,26,27,30,21,25,32,31,28,23,19,22));