Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1302 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22⋊C4 |
| commutator | C2×C4 | C42×C13 |
| Frattini, cosocle | C2×C42 | C22×C13 |
| socle | C22×C13 | #95 |
| prime radical | C23×C13 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
144 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 144 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)(16,18)(17,20)(19,21)(22,23,25,29,27,30,26,35)(24,31,33,37,32,28,34,36);(15,18,20,19)(22,24,26,34,27,32,25,33)(23,28,29,36,30,31,35,37)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 34);