Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#3020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
698 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 698 conjugacy classes of subgroups — too many to draw. Only its 212 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 — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,22)(15,19)(16,28)(17,20)(18,24)(21,31)(23,39)(25,36)(26,37)(27,34)(29,33)(30,35)(32,40)(38,41)(42,43);(12,14)(13,18)(15,23)(16,29)(17,25)(19,32)(20,33)(21,36)(22,28)(24,31)(26,42)(27,41)(30,40)(34,43)(35,38)(37,39);(12,15,17,27)(13,19,20,34)(14,23,25,41)(16,30,31,26)(18,32,33,43)(21,37,28,35)(22,38,36,39)(24,42,29,40);(13,20)(18,33)(19,34)(21,28)(22,36)(32,43)(35,37)(38,39)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1752);