Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #386 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 108 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)(12,13,16,21,17,20,22,35)(14,18,25,32,24,33,29,36)(15,26,28,37,27,19,30,34)(23,31,39,42,38,41,40,43);(12,14,22,25,17,24,16,29)(13,18,21,36,20,33,35,32)(15,23,28,40,27,38,30,39)(19,31,37,42,26,41,34,43);(12,15)(13,19)(14,23)(16,30)(17,27)(18,31)(20,26)(21,37)(22,28)(24,38)(25,40)(29,39)(32,43)(33,41)(34,35)(36,42)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 647);