Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C16 |
| direct product | C3×#72214020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #9837 |
| commutator | C82 | C22×C3×C4 |
| Frattini, cosocle | C2×C82 | C23×C3 |
| prime radical | C3×#491469 | C2 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C82 ◁ G | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,6,10,8,13,14,12)(7,16,15,11,18,9,17,19)(21,25,27,33)(22,29)(23,32)(26,28,34,35)(30,31);(5,9)(6,14)(7,17)(10,11)(12,19)(13,16)(15,18)(21,26)(22,30)(23,32)(25,28)(27,34)(29,31)(33,35);(20,21,22,25,23,28,29,26,24,27,31,33,32,35,30,34)228 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)(4,5,6,10,8,13,14,12)(7,16,15,11,18,9,17,19)(21,25,27,33)(22,29)(23,32)(26,28,34,35)(30,31), (5,9)(6,14)(7,17)(10,11)(12,19)(13,16)(15,18)(21,26)(22,30)(23,32)(25,28)(27,34)(29,31)(33,35), (20,21,22,25,23,28,29,26,24,27,31,33,32,35,30,34));