Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#12780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1790 |
| commutator | C2×Q8 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| prime radical | C7×#3551 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
756 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)(10,20)(11,24,12,17)(13,18)(14,19)(15,21,32,26)(16,25,33,27)(22,36)(23,34)(28,30)(29,35)(31,37,39,38);(8,10)(9,15)(11,21)(12,26)(13,28)(14,23)(16,30)(17,34)(18,31)(19,32)(20,24)(22,29)(25,38)(27,37)(33,36)(35,39);(8,11,14,12)(9,16,19,33)(10,21,23,26)(13,27,29,25)(15,30,32,36)(17,35,24,18)(20,31,34,39)(22,38,28,37)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5768);