Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#13146 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #466 |
| commutator | C2×C8 | C22×C4×C11 |
| Frattini, cosocle | C22×C8 | C23×C11 |
| socle | C2×C11 | #1934 |
| prime radical | C2×C11×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ C11×#405 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,23,19,24,18,41)(14,25,28,53,30,54,29,63)(15,31,34,66,36,48,35,68)(16,37,38,43,40,69,39,22)(20,42,45,62,46,61,52,27)(21,47,49,32,50,67,65,33)(26,55,58,71,60,73,59,44)(51,72,74,56,64,70,75,57);(12,14,16,27,17,28,38,42,19,30,40,62,18,29,39,61)(13,20,43,54,41,52,37,53,24,46,22,25,23,45,69,63)(15,26,32,56,34,58,67,70,36,60,33,57,35,59,47,72)(21,44,31,64,65,73,68,74,50,71,48,51,49,55,66,75);(12,15,16,32,17,34,38,67,19,36,40,33,18,35,39,47)(13,21,22,48,41,65,69,66,24,50,43,31,23,49,37,68)(14,26,27,56,28,58,42,70,30,60,62,57,29,59,61,72)(20,44,25,51,52,73,63,75,46,71,54,64,45,55,53,74)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6134);