Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#20998 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #386 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#386 |
| prime radical | C2×C7×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
1704 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)(8,9,13,18,14,19,12,17)(10,20,23,31,24,30,22,15)(11,25,26,16,28,32,27,33)(21,29,35,38,37,39,36,34);(8,10)(9,15,19,31)(11,21)(12,22)(13,23)(14,24)(16,29,33,39)(17,30,18,20)(25,34,32,38)(26,35)(27,36)(28,37);(8,11)(9,16)(10,21)(12,26)(13,27)(14,28)(15,29)(17,32)(18,25)(19,33)(20,34)(22,35)(23,36)(24,37)(30,38)(31,39);(8,12,14,13)(9,17,19,18)(10,22,24,23)(11,26,28,27)(15,30,31,20)(16,32,33,25)(21,35,37,36)(29,38,39,34)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13986);