Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#1461 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22⋊C4 |
| commutator | C22 | C22×C8×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C22×C13 | C2×C22⋊C4 |
| prime radical | C23×C13 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
248 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,20,18,21,16,19)(22,23)(24,30,32,27)(25,28)(26,29)(31,36,37,34)(33,35);(22,24,26,32)(23,27,29,30)(25,31,33,37)(28,34,35,36);(14,16,18,17)(15,19,21,20)(22,25,26,33)(23,28,29,35)(24,31,32,37)(27,34,30,36)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 193);