Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C9 · Schur multiplier (H2) = C2×C32 |
| direct product | D4×#6903 |
| unambiguous semidirect product | C4⋊(C2×#6903) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C27 | C22×C32 |
| commutator, socle | C2×C3 | C22×C32×C9 |
| Frattini, cosocle | C2×C9 | C22×C33 |
| prime radical | D4×H(3,𝔽3) | C9 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C27 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,3,4)(5,6,10,8,12,22,18,28,43,9,13,23,19,29,44,37,52,66,20,30,45,38,53,67,59,73,81)(7,11,21,16,26,41,35,50,64,17,27,42,36,51,65,58,72,80,15,25,40,34,49,63,57,71,79)(14,24,39,32,47,61,55,69,77,31,46,60,54,68,76,74,82,84,33,48,62,56,70,78,75,83,85);(2,4)(5,7,14)(6,11,24)(8,16,32)(9,17,31)(10,21,39)(12,26,47)(13,27,46)(15,33,20)(18,35,55)(19,36,54)(22,41,61)(23,42,60)(25,48,30)(28,50,69)(29,51,68)(34,56,38)(37,58,74)(40,62,45)(43,64,77)(44,65,76)(49,70,53)(52,72,82)(57,75,59)(63,78,67)(66,80,84)(71,83,73)(79,85,81);(7,15,17)(11,25,27)(14,31,33)(16,34,36)(21,40,42)(24,46,48)(26,49,51)(32,54,56)(35,57,58)(39,60,62)(41,63,65)(47,68,70)(50,71,72)(55,74,75)(61,76,78)(64,79,80)(69,82,83)(77,84,85)495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 2212);