Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11307 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #397 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1443 |
| prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
548 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)(10,14,19,28)(11,21)(12,25)(13,16)(15,23)(17,26)(18,27,32,39)(20,34,33,29)(22,35,37,30)(24,31)(36,38);(8,10)(9,14,16,28)(11,22,23,37)(12,20)(13,19)(15,30)(17,29,25,34)(18,24,32,38)(21,35)(26,33)(27,36)(31,39);(8,11)(9,15,16,21)(10,18,19,32)(12,24)(13,23)(14,27)(17,31,25,36)(20,22,33,37)(26,38)(28,39)(29,30)(34,35)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4295);