Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C2×C13×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C42 |
| commutator | C4 | C2×C42×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| socle | C22×C13 | #97 |
| prime radical | C23×C13 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ G | |
168 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31);(3,4,5,9,6,11,13,17,7,10,12,18,14,8,15,16);(4,8,10,11)(5,12)(9,17,18,16)(13,15)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 841);