Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1593 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C2×D4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C42 | C23×C11 |
| socle, prime radical | C22×C11 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
148 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,14,15,17,19,18)(20,21)(22,28)(23,31)(24,26)(25,29)(27,33)(30,35)(32,34);(13,17)(14,18)(20,22,24,30)(21,25,26,34)(23,32,33,29)(27,35,31,28);(12,14,16,17,15,18,19,13)(21,26)(25,34)(27,31)(28,35)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 325);