Finite Group DatabasebetaFinite Group Databasebeta| order | 344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C7×S4 (order 1008) · Inn(G) = C22 · Out(G) = C2×C3×D3×C7 |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C1 |
| direct product | Q8×C43 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C43 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C43 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C43 ◁ G | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 344 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(1,5,3,7)(2,6,4,8)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(344, 10);