Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#1794 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #120 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #379 |
| prime radical | C11×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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)(12,13,17,21,18,22,16,29)(14,23,25,19,26,34,30,33)(15,20,27,36,28,37,31,35)(24,39,42,43,41,38,40,32);(12,14)(13,19)(15,24)(16,30)(17,25)(18,26)(20,32)(21,34)(22,33)(23,29)(27,42)(28,41)(31,40)(35,38)(36,39)(37,43);(12,15)(13,20)(14,24)(16,31)(17,27)(18,28)(19,32)(21,36)(22,37)(23,38)(25,42)(26,41)(29,35)(30,40)(33,43)(34,39)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 526);