Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C23×C11 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
202 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,18,33)(14,23,25,40)(15,28,34,30)(16,21,17,22)(19,35,36,24)(20,32,43,31)(26,39,27,41)(29,37,42,38);(12,14,16,26)(13,19,21,37)(15,29,31,24)(17,27,18,25)(20,39,28,23)(22,38,33,36)(30,40,43,41)(32,35,34,42);(12,13,17,22)(14,19,27,38)(15,30,32,20)(16,21,18,33)(23,24,41,42)(25,36,26,37)(28,31,43,34)(29,40,35,39)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 349);