Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#33385 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C22×D8 |
| commutator | C8 | C23×C4×C7 |
| Frattini, cosocle | C2×C8 | C24×C7 |
| socle | C2×C7 | C22×#102 |
| prime radical | C7×#3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×#41 ◁ C2×C7×#131 ◁ G | |
1114 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)(8,9,13,19,15,20,14,35)(10,21,24,47,25,48,36,44)(11,17,28,43,26,41,29,58)(12,30,32,45,34,60,33,18)(16,31,38,52,40,61,39,23)(22,46,51,68,49,67,59,66)(27,54,56,65,53,70,57,42)(37,55,63,69,62,71,64,50);(8,10)(9,16)(11,22)(12,31)(13,24)(14,36)(15,25)(17,37)(18,44)(19,38)(20,40)(21,30)(23,33)(26,49)(27,55)(28,51)(29,59)(32,52)(34,61)(35,39)(41,62)(42,66)(43,63)(45,47)(46,54)(48,60)(50,57)(53,71)(56,69)(58,64)(65,68)(67,70);(8,11)(9,17)(10,22)(12,27)(13,28)(14,29)(15,26)(16,37)(18,42)(19,43)(20,41)(21,46)(23,50)(24,51)(25,49)(30,54)(31,55)(32,56)(33,57)(34,53)(35,58)(36,59)(38,63)(39,64)(40,62)(44,66)(45,65)(47,68)(48,67)(52,69)(60,70)(61,71);(11,26)(17,41)(22,49)(27,53)(28,29)(37,62)(42,65)(43,58)(46,67)(50,69)(51,59)(54,70)(55,71)(56,57)(63,64)(66,68)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26373);