Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#22457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| prime radical | C2×C7×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 71 points:
(1,2,3,4,5,6,7)(8,9)(10,20)(11,25,28,16)(12,17)(13,32)(14,19)(15,23)(18,35)(21,48,51,46)(22,47)(24,44)(26,55,59,39)(27,40,62,56)(29,63)(30,45)(31,43)(33,38)(34,42)(36,50,54,60)(37,53)(41,52)(49,65,69,68)(57,64,70,71)(58,66,61,67);(8,10,14,24)(9,15,19,38)(11,21,28,51)(12,29,31,53)(13,33,35,23)(16,36,25,54)(17,41,43,47)(18,44,32,20)(22,34,52,30)(26,57,59,70)(27,60,62,50)(37,45,63,42)(39,65,55,68)(40,48,56,46)(49,61,69,58)(64,67,71,66);(8,11)(9,16)(10,21)(12,26)(13,27)(14,28)(15,36)(17,39)(18,40)(19,25)(20,46)(22,49)(23,50)(24,51)(29,57)(30,58)(31,59)(32,56)(33,60)(34,61)(35,62)(37,64)(38,54)(41,65)(42,66)(43,55)(44,48)(45,67)(47,68)(52,69)(53,70)(63,71);(8,12,14,31)(9,17,19,43)(10,22,24,52)(11,26,28,59)(13,34,35,30)(15,37,38,63)(16,39,25,55)(18,45,32,42)(20,47,44,41)(21,49,51,69)(23,53,33,29)(27,61,62,58)(36,64,54,71)(40,67,56,66)(46,68,48,65)(50,70,60,57)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 15445);