Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#1270 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22⋊C4 |
| commutator | C2×C4 | C2×C8×C13 |
| Frattini, cosocle | C2×C42 | C22×C13 |
| prime radical | C22×D4×C13 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C42×C13 ◁ G | |
202 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20,21,18)(17,19)(22,23)(24,26)(25,28)(27,29)(30,31,32,34,33,35,36,37);(14,16)(15,18)(17,21)(19,20)(23,26)(24,29)(27,28)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2);