Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11313 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #388 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1443 |
| prime radical | C24×C7 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
532 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)(10,19,22,15)(11,23)(12,27)(13,18)(14,17)(16,25)(20,37,39,33)(21,31,28,32)(24,30,38,36)(26,35)(29,34);(8,10,12,21,14,22,13,28)(9,15,27,32,17,19,18,31)(11,24,25,20,26,38,29,39)(16,33,35,30,34,37,23,36);(8,11)(9,16,17,34)(10,20)(12,25)(13,29)(14,26)(15,30,19,36)(18,23,27,35)(21,38)(22,39)(24,28)(31,33,32,37)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4301);