Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C11×#353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #99 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C2×D4 | C23×C11 |
| prime radical | C2×C11×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C23×C11 ◁ C22×D4×C11 ◁ G | |
368 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21);(3,4)(5,8,10,7)(6,9);(4,7)(5,10)(8,9)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 856);