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) = C23 |
| direct product | C2×C13×#346 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C13 | C22 |
| commutator | C2 | C2×C4×C8×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| socle, prime radical | C23×C13 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C13 ◁ G | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35);(7,8,9,13,10,14,15,20,11,12,16,18,17,19,21,22);(3,4,5,6)(8,12)(13,18)(14,19)(20,22)1040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 838);