Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×#1572 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C2×D4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C42 | C23×C11 |
| socle | C22×C11 | C2×C4⋊C4 |
| prime radical | C23×C11 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C2×C42×C11 ◁ G | |
150 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 150 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) = 704 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,23)(22,26)(24,27)(28,29,31,34,32,35,38,41)(30,33,37,40,36,39,42,43);(12,14,16,23,17,24,15,22)(13,18,19,21,20,26,25,27)(28,30)(29,33)(31,37)(32,36)(34,40)(35,39)(38,42)(41,43);(12,15,17,16)(13,19,20,25)(14,22,24,23)(18,21,26,27)(30,36)(33,39)(37,42)(40,43)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 304);