Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C11×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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)(12,13,19,25)(14,23,28,34)(15,29,31,42)(16,24,18,22)(17,20,27,35)(21,36,38,26)(30,43,33,39)(32,40,41,37);(12,14,18,27,19,28,16,17)(13,20,24,34,25,35,22,23)(15,26,33,41,31,36,30,32)(21,29,40,43,38,42,37,39);(12,15)(13,21)(14,26)(16,30)(17,32)(18,33)(19,31)(20,29)(22,37)(23,39)(24,40)(25,38)(27,41)(28,36)(34,43)(35,42)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 946);