Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2380 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#3153 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
706 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)(10,14,20,30)(11,22,25,32)(12,18)(13,17)(15,26,33,35)(16,24)(19,31)(21,29,27,34)(23,28)(36,39)(37,38);(8,10)(9,14)(11,23)(12,20)(13,27)(15,31)(16,29)(17,34)(18,30)(19,35)(21,24)(22,28)(25,37)(26,36)(32,38)(33,39);(8,11,24,35,12,25,13,26)(9,15,17,22,18,33,16,32)(10,19,27,23,20,36,21,37)(14,28,29,31,30,38,34,39);(9,16,18,17)(10,20)(14,29,30,34)(15,32,33,22)(19,36)(21,27)(23,37)(28,31,38,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16350);