Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#12569 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #102 |
| commutator | C8 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C2×#102 |
| prime radical | C24×C7 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ C23×C7×C8 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,23)(15,20)(16,30)(17,33)(18,22)(19,25)(21,31)(24,38)(26,36)(27,42)(28,35)(29,37)(32,43)(34,40)(39,41);(12,14,16,26,15,24,29,41,18,28,32,25,17,27,31,40)(13,19,21,35,33,39,43,38,22,36,37,23,20,34,30,42);(12,15,18,17)(13,20,22,33)(14,24,28,27)(16,29,32,31)(19,34,36,39)(21,30,37,43)(23,38,35,42)(25,40,26,41)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5557);