Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C16 · Schur multiplier (H2) = C22 |
| direct product | C7×#13628 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C16 | C23 |
| commutator | C22 | C22×C7×C16 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C22×C7 | C22×C16 |
| prime radical | C23×C7 | C2×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C16 ◁ G | |
206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 75 points:
(1,2,3,4,5,6,7)(9,11)(12,13,15,20,16,21,28,38,17,22,29,39,31,41,50,60,18,23,30,40,32,42,51,61,33,43,52,62,53,63,68,73)(14,19,25,35,26,36,47,57,27,37,48,58,49,59,67,72,24,34,44,54,45,55,64,69,46,56,65,70,66,71,74,75);(8,9,10,11)(12,14)(13,19)(15,25)(16,26)(17,27)(18,24)(20,35)(21,36)(22,37)(23,34)(28,47)(29,48)(30,44)(31,49)(32,45)(33,46)(38,57)(39,58)(40,54)(41,59)(42,55)(43,56)(50,67)(51,64)(52,65)(53,66)(60,72)(61,69)(62,70)(63,71)(68,74)(73,75);(14,24)(19,34)(25,44)(26,45)(27,46)(35,54)(36,55)(37,56)(47,64)(48,65)(49,66)(57,69)(58,70)(59,71)(67,74)(72,75)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6616);