Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C5×#41 (order 320) · Inn(G) = C22 · Out(G) = C24×C5 |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C22 |
| commutator | C2 | C2×C8×C11 |
| Frattini, cosocle | C8 | C22×C11 |
| socle | C2×C11 | C2×C8 |
| prime radical | C22×C11 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ G | |
26 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)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27);(13,21)(15,23)(17,25)(19,27)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 59);