Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×#1309 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #95 |
| commutator | C23 | C42×C11 |
| Frattini, cosocle | C2×C42 | C22×C11 |
| prime radical | C23×C11 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C42×C11 ◁ G | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,17,15,16,18,19)(21,24)(22,26);(13,16)(14,18)(20,21,22,25,23,24,26,27)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 41);