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) = C22×C4 |
| direct product | C11×#2126 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #99 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C2×D4 | C23×C11 |
| socle | C2×C11 | C2×#99 |
| prime radical | C11×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C11×#41 ◁ G | |
258 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 258 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) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18)(15,19,23,17)(16,20)(21,25,27,24)(22,26);(13,17)(15,23)(18,24)(19,20)(21,27)(25,26);(12,14,16,22)(13,18,20,26)(15,21,23,27)(17,24,19,25)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 858);