Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×C13×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C13 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31);(3,4,7,10)(5,12,13,9)(6,14,16,18)(8,11,17,15);(3,5,7,13)(4,9,10,12)(6,15,16,11)(8,14,17,18);(3,6)(4,5)(7,16)(8,9)(10,13)(11,14)(12,17)(15,18);(3,5)(4,6)(7,13)(8,14)(9,11)(10,16)(12,15)(17,18)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2258);