Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C11 · Schur multiplier (H2) = C22 |
| direct product | C2×C11×#1398 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | C42 |
| commutator | C4 | C2×C4×C8×C11 |
| Frattini, cosocle | C2×C16 | C23×C11 |
| socle | C22×C11 | #346 |
| prime radical | C23×C11 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45);(3,4,5,10,6,12,15,25,7,13,16,23,17,27,29,33,8,11,14,24,18,26,28,34,19,9,20,21,30,22,31,32);(4,9,11,13)(5,14)(10,23,24,21)(12,22,26,27)(15,28)(16,20)(25,33,34,32)(29,31)968 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6538);