Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2170 |
| subgroup | G/H | |
|---|---|---|
| center | C11×C16 | D4 |
| commutator | C4 | C22×C8×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | C8×D4 |
| prime radical | C11×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C11×C16 ◁ C2×C11×C16 ◁ G | |
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 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)(12,13,17,21,16,22,31,39,18,23,33,40,32,20,36,38)(14,24,26,29,27,35,41,30,28,37,42,43,25,19,34,15);(12,14)(13,19)(15,21)(16,27)(17,26)(18,28)(20,37)(22,24)(23,35)(25,32)(29,39)(30,40)(31,41)(33,42)(34,36)(38,43);(12,15,17,24,16,29,31,35,18,30,33,37,32,43,36,19)(13,14,21,26,22,27,39,41,23,28,40,42,20,25,38,34)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 902);