Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#23753 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#400 |
| prime radical | C24×C7 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
648 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)(8,9,12,17,13,18,14,29)(10,19,21,31,23,34,22,15)(11,24,27,38,26,16,28,33)(20,32,36,39,35,37,25,30);(8,10,14,21,13,23,12,22)(9,15,29,19,18,31,17,34)(11,25,28,20,26,36,27,35)(16,32,38,39,24,37,33,30);(8,11,12,27,13,26,14,28)(9,16,17,33,18,24,29,38)(10,20,21,36,23,35,22,25)(15,30,19,32,31,39,34,37);(11,26)(16,24)(20,35)(25,36)(27,28)(30,39)(32,37)(33,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16741);