Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C16 |
| direct product | C3×#72213956 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#1712 |
| commutator, Frattini, cosocle | C82 | C24×C3 |
| LCS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C82 ◁ G | |
2414 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)(4,5)(6,9)(7,16)(8,10)(11,13)(12,14)(15,17)(18,19)(21,25)(22,28)(23,30)(26,27)(29,34)(31,35)(32,33);(5,9)(6,13)(7,14)(10,17)(11,12)(15,19)(16,18)(21,26)(22,28)(23,30)(25,27)(29,33)(31,35)(32,34);(5,10)(9,17)(11,18)(12,16)(21,27)(25,26)(29,32)(33,34);(20,21,28,33,23,29,31,26,24,27,35,34,30,32,22,25)372 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)(4,5)(6,9)(7,16)(8,10)(11,13)(12,14)(15,17)(18,19)(21,25)(22,28)(23,30)(26,27)(29,34)(31,35)(32,33), (5,9)(6,13)(7,14)(10,17)(11,12)(15,19)(16,18)(21,26)(22,28)(23,30)(25,27)(29,33)(31,35)(32,34), (5,10)(9,17)(11,18)(12,16)(21,27)(25,26)(29,32)(33,34), (20,21,28,33,23,29,31,26,24,27,35,34,30,32,22,25));