Finite Group DatabasebetaFinite Group Databasebeta| order | 1372 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C72 · Schur multiplier (H2) = C2 |
| direct product | C22×7−1+2 |
| unambiguous semidirect product | (C22×C49)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C72 |
| commutator, Frattini, cosocle | C7 | C22×C72 |
| prime radical | C22×C72 | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 196 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2)(5,6,8,12,19,30,39,7,10,14,21,32,41,48,11,17,24,34,43,40,49,18,28,23,36,31,42,50,29,27,26,20,33,44,51,38,16,13,22,35,45,52,47,9,15,25,37,46,53);(3,4)(6,9,16,27,28,17,10)(8,13,23,14,15,26,24)(12,20,21,22,34,25,36)(19,31,37,43,35,32,33)(30,40,44,46,41,42,45)(39,48,49,50,51,52,53)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1372, 25);