Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×#99 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C2×Q8 | C23×C11 |
| prime radical | C2×C11×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C23×C11 ◁ C22×Q8×C11 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 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) = 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)(12,13,16,20,17,21,32,39)(14,22,25,40,23,36,30,42)(15,26,28,38,29,19,31,37)(18,33,35,27,34,41,43,24);(12,14)(13,18)(15,27)(16,30)(17,23)(19,36)(20,35)(21,34)(22,26)(24,29)(25,32)(28,33)(31,41)(37,42)(38,40)(39,43);(14,23)(18,34)(22,36)(24,27)(25,30)(33,41)(35,43)(40,42)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 865);