Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#1868 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #120 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #385 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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)(12,13,16,20,17,21,27,33)(14,18,24,30,23,29,37,41)(15,19,26,32,25,31,38,42)(22,34,36,43,35,28,39,40);(12,14,16,24,17,23,27,37)(13,18,20,30,21,29,33,41)(15,22,26,36,25,35,38,39)(19,28,32,40,31,34,42,43);(12,15)(13,19)(14,22)(16,26)(17,25)(18,28)(20,32)(21,31)(23,35)(24,36)(27,38)(29,34)(30,40)(33,42)(37,39)(41,43)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 600);