Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11264 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #124 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #400 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(9,10)(12,13,17,21,18,23,16,22)(14,24,26,19,28,35,27,36)(15,29,31,39,32,38,33,20)(25,37,42,43,30,34,41,40);(8,9,11,10)(12,14,16,27,18,28,17,26)(13,19,22,24,23,36,21,35)(15,30,31,41,32,25,33,42)(20,37,29,43,39,34,38,40);(8,10,11,9)(12,15,16,31,18,32,17,33)(13,20,21,38,23,39,22,29)(14,25,26,41,28,30,27,42)(19,34,24,40,36,37,35,43)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4252);