Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#3291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ G | |
460 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,18,23)(14,24,25,26)(15,20,19,27)(16,22,17,21);(12,14)(13,19)(15,23)(16,27)(17,20)(18,25)(21,26)(22,24);(12,15)(13,14)(16,26)(17,24)(18,19)(20,21)(22,27)(23,25);(12,15)(13,20)(14,21)(16,24)(17,26)(18,19)(22,25)(23,27)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2023);