Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1359 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #331 |
| commutator | C42 | C2×C4×C11 |
| Frattini, cosocle | C4×C8 | C22×C11 |
| prime radical | C11×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C4×C8×C11 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,19,22,17,21,25,16,20,24,27,15,18,23,26);(13,17)(14,21)(15,22)(18,20)(19,27)(23,24)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 91);