Finite Group DatabasebetaFinite Group Databasebeta| order | 1372 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C72 · Schur multiplier (H2) = C72 |
| direct product | C4×H(3,𝔽7) |
| unambiguous semidirect product | (C4×C72)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C72 |
| commutator | C7 | C4×C72 |
| Frattini, socle, cosocle | C2×C7 | C2×C72 |
| prime radical | C2×H(3,𝔽7) | C2 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ G | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 196 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,3,4)(5,6,8,12,19,30,39)(7,10,14,21,32,41,48)(9,15,25,37,46,53,47)(11,17,24,34,43,40,49)(13,22,35,45,52,38,16)(18,28,23,36,31,42,50)(20,33,44,51,29,27,26);(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, 12);