Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C13 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C13×#431 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C13 | C22 |
| commutator | C2 | C23×C8×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C23×C13 | C22×C4 |
| prime radical | C24×C13 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C13 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31);(3,4)(5,8)(6,7)(9,10)(11,12,13,15,14,16,17,18);(4,7)(8,10);(3,5,6,9)(4,8,7,10)1040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1649);