Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1375 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C42 |
| commutator | C4 | C4×C8×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| socle | C22×C13 | #97 |
| prime radical | C23×C13 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ G | |
140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,21,26,22,24,30,31,23,27,29,33,20,25,28,32);(15,17)(19,24,27,25)(21,29)(26,32,33,31)(28,30)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 107);