Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #476 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C23×C7 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
294 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)(9,11)(12,13)(14,24,28,42)(15,30)(16,20)(17,35)(18,23)(19,26,37,25)(21,32)(22,31)(27,40,29,41)(33,38,43,36)(34,39);(12,14,17,27,18,28,31,29)(13,19,22,36,23,37,35,38)(15,25,16,33,32,26,34,43)(20,24,30,41,39,42,21,40);(8,9,10,11)(12,15,31,34,18,32,17,16)(13,20,35,21,23,39,22,30)(14,25,29,43,28,26,27,33)(19,24,38,40,37,42,36,41)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6175);