Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#479908 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×C42 |
| commutator | C4 | C22×C3×C4×C8 |
| Frattini, cosocle | C4×C8 | C24×C3 |
| socle | C22×C3 | C22×#97 |
| prime radical | C2×C3×#41 | C42 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
1268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 43 points:
(9,10)(11,16)(12,17)(13,18)(14,15)(19,28)(20,25)(21,29)(22,26)(23,30)(24,27)(31,36)(32,39)(33,37)(34,40)(35,38)(41,42,43);(1,2,3,5,4,6,7,8)(11,19,20,31)(12,22)(16,28,25,36)(17,26)(21,34,33,32)(23,35)(29,40,37,39)(30,38);(10,15)(16,25)(17,26)(18,27)(28,36)(29,37)(30,38)(39,40);(9,11,12,21,13,19,23,32,14,20,22,33,24,31,35,34)(10,16,17,29,18,28,30,39,15,25,26,37,27,36,38,40)492 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((9,10)(11,16)(12,17)(13,18)(14,15)(19,28)(20,25)(21,29)(22,26)(23,30)(24,27)(31,36)(32,39)(33,37)(34,40)(35,38)(41,42,43), (1,2,3,5,4,6,7,8)(11,19,20,31)(12,22)(16,28,25,36)(17,26)(21,34,33,32)(23,35)(29,40,37,39)(30,38), (10,15)(16,25)(17,26)(18,27)(28,36)(29,37)(30,38)(39,40), (9,11,12,21,13,19,23,32,14,20,22,33,24,31,35,34)(10,16,17,29,18,28,30,39,15,25,26,37,27,36,38,40));