Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#2132 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #99 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×Q8 | C23×C13 |
| socle | C2×C13 | C2×#99 |
| prime radical | C13×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
250 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,23,19,24,16,21)(17,22,25,29,26,20,27,28);(15,20)(17,26)(21,28)(22,24)(23,29)(25,27);(14,16,19,18)(15,21,24,23)(17,25,26,27)(20,28,22,29)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 864);