Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1696 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C11 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
140 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 140 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) = 1408 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,19)(14,20,22,27)(15,24,21,18)(17,25,26,23)(28,29,32,35)(30,36,38,43)(31,39,41,34)(33,40,42,37);(12,14,21,25,16,22,15,23)(13,17,24,27,19,26,18,20);(14,22)(17,26)(20,27)(23,25)(28,30,32,38)(29,33,35,42)(31,40,41,37)(34,43,39,36)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 428);