Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#2199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C11 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C11 ◁ C2×D4×C11 ◁ G | |
270 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 56 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)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27);(12,14)(13,17)(15,22)(16,21)(18,20)(19,24)(23,27)(25,26);(14,21)(15,23)(17,24)(20,26)187 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 931);