Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#12666 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C42 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C22×C7 | C2×#97 |
| prime radical | C23×C7 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,13,16,21,15,20,24,27,17,22,25,14,19,18,26,23);(9,11)(13,18,22,20)(14,23,21,27)(16,25)(24,26);(12,14,24,13,19,27,16,18,17,21,26,22,15,23,25,20)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5654);