Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1913 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #386 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C24 | C23×C11 |
| prime radical | C11×#534 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C24×C11 ◁ G | |
380 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 380 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) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17,21,25)(15,22,23,18)(16,19)(20,24)(26,27);(12,14)(13,17,19,25)(15,20)(16,21)(18,24,22,27)(23,26);(12,15)(13,18)(14,20)(16,23)(17,24)(19,22)(21,26)(25,27)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 645);