Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #99 |
| commutator | C22×C4 | C2×C4×C11 |
| Frattini, cosocle | C2×C4⋊C4 | C22×C11 |
| prime radical | C23×C11 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C23×C11 ◁ C2×C11×C4⋊C4 ◁ G | |
116 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,16,17,18)(20,21,23,27,24,26,30,33)(22,25,29,31,28,32,35,34);(12,13)(14,18)(15,17)(16,19)(20,22)(21,25,26,32)(23,29)(24,28)(27,34,33,31)(30,35)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 85);