Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C13 · Schur multiplier (H2) = C2×C42 |
| direct product | C13×#1744 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C13 | C22 |
| commutator | C2 | C2×C4×C8×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle, prime radical | C23×C13 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C13 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19)(20,23)(21,22)(24,25)(26,27,28,30,29,31,32,33);(14,15,16,17)(19,22)(23,25);(18,20,21,24)(19,23,22,25)1040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 476);