Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#7438 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #353 |
| commutator | C2×M16 | C2×C4×C7 |
| Frattini, cosocle | #408 | C22×C7 |
| socle | C2×C7 | #1352 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C7×#408 ◁ G | |
174 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 2 generators on 71 points:
(1,2,3,4,5,6,7)(8,9,12,17)(10,20,22,46)(11,16,26,39)(13,19,31,44)(14,18,30,45)(15,33,34,54)(21,35,49,42)(23,47,52,69)(24,37,29,58)(25,36,53,62)(27,41,56,43)(28,40,55,67)(32,38,60,66)(48,64,59,71)(50,63,51,68)(57,65,61,70);(8,10)(9,15)(11,25,27,54)(12,29,30,59)(13,24)(14,23)(16,37,41,64)(17,42,45,68)(18,36)(19,35)(20,40,47,65)(21,28,50,57)(22,31,52,60)(26,49)(32,48)(33,61)(34,44,62,66)(38,63)(39,46)(43,69)(51,56)(53,55)(58,67)(70,71)217 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 426);