Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1638 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C11 |
| prime radical | C2×D4×C11 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
194 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 704 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,20)(14,21,23,26)(15,19,16,24)(18,25,27,22)(28,29,33,37)(30,38,41,42)(31,36,32,35)(34,40,43,39);(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24)(28,30,33,41)(29,34,37,43)(31,40,32,39)(35,42,36,38);(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21)(29,35)(31,32)(34,42)(36,37)(38,43)(39,40)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 370);