Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#12839 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #427 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1808 |
| prime radical | C2×C7×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×M16 ◁ C2×C7×#131 ◁ G | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 71 points:
(1,2,3,4,5,6,7)(8,9)(10,20)(11,26,28,16)(12,30)(13,33)(14,18)(15,23)(17,21)(19,31)(22,47,49,40)(24,53)(25,42)(27,57,59,60)(29,38,58,56)(32,61)(34,48,54,70)(35,52)(36,63)(37,45,55,67)(39,46)(41,44)(43,51)(50,65,69,68)(62,71,64,66);(8,10,21,44,13,24,32,52,14,25,46,63,31,51,12,23)(9,15,30,43,19,36,39,42,18,35,61,53,33,41,17,20)(11,22,45,66,29,50,59,70,28,49,67,71,58,69,27,48)(16,34,60,65,38,62,55,47,26,54,57,68,56,64,37,40);(8,11)(9,16)(10,22)(12,27)(13,29)(14,28)(15,34)(17,37)(18,26)(19,38)(20,40)(21,45)(23,48)(24,50)(25,49)(30,60)(31,58)(32,59)(33,56)(35,54)(36,62)(39,55)(41,64)(42,47)(43,65)(44,66)(46,67)(51,69)(52,70)(53,68)(57,61)(63,71)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5827);