Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C5×#120 (order 640) · Inn(G) = C22 · Out(G) = C22×C5×D4 |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×C4⋊C8 |
| unambiguous semidirect product | C4⋊(C8×C11) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C22 |
| commutator | C2 | C2×C8×C11 |
| Frattini, cosocle | C2×C4 | C22×C11 |
| socle, prime radical | C22×C11 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,4)(5,6,7,9,8,10,11,12)(13,14,15,16,17,18,19,20,21,22,23);(1,2,3,4)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 54);