Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C16 · Schur multiplier (H2) = C22 |
| direct product | C7×#7469 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C16 | C22⋊C4 |
| commutator | C22 | C4×C7×C16 |
| Frattini, cosocle | C22×C16 | C22×C7 |
| socle | C2×C7 | #1374 |
| prime radical | C23×C7 | C2×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C16 ◁ C22×C7×C16 ◁ G | |
174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,11,16,12,18,25,36,13,19,26,37,28,40,49,60,14,17,27,35,29,38,50,58,30,39,51,59,52,61,67,70)(10,20,22,41,23,42,46,62,24,43,47,63,48,64,66,71,21,15,31,32,44,33,53,55,45,34,54,56,65,57,68,69);(8,10)(9,15,17,20)(11,22)(12,23)(13,24)(14,21)(16,32,35,41)(18,33,38,42)(19,34,39,43)(25,46)(26,47)(27,31)(28,48)(29,44)(30,45)(36,55,58,62)(37,56,59,63)(40,57,61,64)(49,66)(50,53)(51,54)(52,65)(60,69,70,71)(67,68)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 457);