Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C23×C5×S4 (order 960) · Inn(G) = C22 · Out(G) = C23×C5×D3 |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C22 |
| commutator | C2 | C22×C4×C11 |
| Frattini, cosocle | C4 | C23×C11 |
| socle | C2×C11 | C22×C4 |
| prime radical | C11×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,19,20,21,22,23,24,25,26,27);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 166);