Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#1835 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C2×D4 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C42×C13 ◁ G | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(18,19)(20,23)(21,22)(24,25)(26,27)(28,36)(29,32)(30,40)(31,35)(33,38)(34,37)(39,41);(19,22)(23,25)(26,28,37,39,31,38,30,29)(27,32,40,33,35,41,34,36);(18,20,21,24)(19,23,22,25)(26,29,30,38,31,39,37,28)(27,33,34,32,35,36,40,41)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 567);