Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C11 |
| prime radical | C11×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C11 ◁ C2×Q8×C11 ◁ G | |
216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 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)(14,23)(15,27)(16,19)(17,22)(18,24)(20,30)(21,32)(25,40)(26,38)(28,36)(29,37)(31,41)(33,39)(34,42)(35,43);(12,14,17,26)(13,18,22,36)(15,28,30,24)(16,31,32,25)(19,34,21,35)(20,38,27,23)(29,42,33,43)(37,40,39,41);(13,19)(16,32)(18,34)(20,39)(21,22)(23,40)(25,31)(27,37)(29,33)(35,36)(38,41)(42,43)187 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 935);