Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#12770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1790 |
| commutator | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| prime radical | C7×#3551 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
752 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)(8,9,12,17,13,15,25,30)(10,18,21,34,20,35,38,39)(11,22,24,33,23,16,26,31)(14,27,29,19,28,36,32,37);(8,10,13,20)(9,14,15,28)(11,19,23,37)(12,21,25,38)(16,18,22,35)(17,32,30,29)(24,36,26,27)(31,39,33,34);(9,15)(12,25)(14,28)(21,38)(24,26)(27,36)(31,33)(34,39)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5758);