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) = C2 |
| direct product | C7×#11413 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C42 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×#97 |
| prime radical | C23×C7 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ G | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9,10,13,11,12,14,15)(17,21,22,27)(18,24)(23,30,29,28)(25,31);(16,17,18,23,19,21,25,28,20,22,24,29,26,27,31,30);(9,12)(13,15)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4401);