Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C4×Q8×C11 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,17)(18,19)(20,21,24,27)(22,25,23,26);(12,13,16,17)(14,18,15,19)(20,22,24,23)(21,25,27,26);(12,14,16,15)(13,18,17,19)(21,26)(22,23)(25,27);(12,15,16,14)(13,18,17,19)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2131);