Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#12012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C11 | D4 |
| commutator | C4 | C22×C11×C16 |
| Frattini, cosocle | C4×C8 | C23×C11 |
| socle | C22×C11 | C2×C4⋊C8 |
| prime radical | C23×C11 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C11 ◁ C2×C4×C8×C11 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,23)(22,26)(24,27)(28,29,30,33,31,34,36,39,32,35,37,40,38,41,42,43);(12,14,16,23,17,24,15,22)(13,18,19,21,20,26,25,27);(12,15,17,16)(13,19,20,25)(14,22,24,23)(18,21,26,27)1232 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 5000);