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) = C22×C4 |
| direct product | C11×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×#99 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C2×D4 | 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 | |
238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 238 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)(14,22,25,37)(15,19)(16,29,31,20)(17,21)(18,24)(23,38,26,40)(27,42,43,35)(28,36)(30,33)(32,39)(34,41);(12,14,17,25)(13,18,21,33)(15,26,28,23)(16,30,31,24)(19,34,36,32)(20,37,29,22)(27,39,43,41)(35,38,42,40);(12,15,17,28)(13,19,21,36)(14,23,25,26)(16,27,31,43)(18,32,33,34)(20,35,29,42)(22,38,37,40)(24,39,30,41)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 861);