Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13027 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #459 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1879 |
| prime radical | C7×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,20,31,38,15,21,11,28,30,33,12,19)(10,22,25,35,26,34,24,36,27,39,23,17,32,37,29,16);(8,10)(9,16,21,36)(11,29)(12,23)(13,24)(14,32)(15,27)(17,33,35,20)(18,34,28,37)(19,39,38,22)(25,31)(26,30);(8,10)(9,17)(11,29)(12,23)(13,24)(14,32)(15,27)(16,33)(18,22)(19,34)(20,36)(21,35)(25,31)(26,30)(28,39)(37,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6015);