Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#2015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C2×C42×C11 ◁ G | |
390 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 704 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(20,21,25,29)(22,30,33,34)(23,28,24,27)(26,32,35,31);(13,15)(16,17)(18,19)(20,22,25,33)(21,26,29,35)(23,32,24,31)(27,34,28,30);(17,19)(21,27)(23,24)(26,34)(28,29)(30,35)(31,32)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 747);