Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#33428 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #117 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C2×C8 | C24×C7 |
| socle | C2×C7 | C2×#432 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C7×#405 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,17,12,18,24,35,13,16,25,31,26,32,38,39)(10,19,21,33,22,29,37,34,20,14,23,27,36,15,28,30);(8,10)(9,14)(11,23)(12,22)(13,20)(15,18)(16,19)(17,33)(21,25)(24,28)(26,36)(27,31)(29,32)(30,39)(34,35)(37,38);(8,10)(9,15,16,29)(11,23)(12,22)(13,20)(14,18,19,32)(17,34,31,30)(21,25)(24,28)(26,36)(27,39,33,35)(37,38);(10,20)(14,19)(15,29)(21,23)(22,36)(27,33)(28,37)(30,34)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26416);