Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1500 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C22×C11 | C2×C22⋊C4 |
| prime radical | C11×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C2×C42×C11 ◁ G | |
238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 238 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) = 35 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,16,17,18)(20,21)(22,28,30,25)(23,26)(24,27)(29,34,35,32)(31,33);(12,13)(14,18)(15,17)(16,19)(20,22)(21,25)(23,29)(24,30)(26,32)(27,28)(31,35)(33,34);(20,23,24,31)(21,26,27,33)(22,29,30,35)(25,32,28,34)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 232);