Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#1454 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C13 | C23 |
| commutator | C22 | C2×C42×C13 |
| Frattini, cosocle | C42 | C23×C13 |
| socle | C22×C13 | C2×C42 |
| prime radical | C23×C13 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C13 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,21,18,20,24,27)(16,19,23,25,22,26,28,29)(30,31)(32,35)(33,37)(34,36)(38,42)(39,41)(40,43)(44,45);(14,16)(15,19)(17,23)(18,22)(20,26)(21,25)(24,28)(27,29)(30,32,33,38,34,39,40,44)(31,35,37,42,36,41,43,45);(15,20)(16,22)(21,27)(23,28)(31,36)(35,41)(37,43)(42,45)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 186);