Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#21783 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2308 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C7×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
1230 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)(10,19,22,14)(11,23,25,16)(12,18)(13,17)(15,21,31,26)(20,28)(24,37,39,32)(27,29)(30,34)(33,35)(36,38);(8,10)(9,14)(11,20)(12,26)(13,22)(15,29)(16,28)(17,19)(18,31)(21,27)(23,33)(24,38)(25,35)(30,37)(32,36)(34,39);(8,10)(9,15)(11,20)(12,26)(13,22)(14,18)(16,30)(17,31)(19,29)(21,27)(23,36)(24,38)(25,35)(28,32)(33,37)(34,39);(8,11,13,25)(9,16,17,23)(10,20,22,35)(12,24,27,39)(14,28,19,33)(15,30,31,36)(18,32,29,37)(21,34,26,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14771);