Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
216 conjugacy classes of subgroups.
| 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)(12,13,17,19,16,20,15,24)(14,21,25,26,22,27,23,18);(12,14,15,23,16,22,17,25)(13,18,19,21,20,26,24,27);(14,22)(15,17)(19,24)(21,27)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 801);