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×#2135 |
| 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 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C11×#41 ◁ 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)(12,13,17,22,18,23,15,20)(14,24,27,42,28,39,25,41)(16,30,29,40,32,21,33,38)(19,34,35,31,36,43,37,26);(12,14,18,28)(13,19,23,36)(15,27,17,25)(16,31,32,26)(20,37,22,35)(21,39,30,24)(29,34,33,43)(38,41,40,42);(12,15,18,17)(13,20,23,22)(14,25,28,27)(16,29,32,33)(19,35,36,37)(21,38,30,40)(24,41,39,42)(26,34,31,43)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 867);