Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S5 |
| commutator, cosocle | C2×SL(2,𝔽5) | C23 |
| Fitting, hypercenter | C22×C4 | S5 |
| hypocenter | SL(2,𝔽5) | #41 |
| LCS | G ▷ C2×SL(2,𝔽5) ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
424 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Permutation generators:
(1,3)(2,4)(5,7,26,17,6,8,25,18)(9,11,42,31,10,12,41,32)(13,40,35,43,14,39,36,44)(15,19,29,33,16,20,30,34)(21,38,27,24,22,37,28,23)(45,48)(46,47)(49,52,70,62,50,51,69,61)(53,56,86,76,54,55,85,75)(57,83,79,88,58,84,80,87)(59,64,73,78,60,63,74,77)(65,81,71,67,66,82,72,68);(1,3,2,4)(5,17,32,15,10)(6,18,31,16,9)(7,20,34,25,14)(8,19,33,26,13)(11,36,42,37,22)(12,35,41,38,21)(23,30,28,43,39)(24,29,27,44,40)(45,47,46,48)(49,61,76,59,54)(50,62,75,60,53)(51,64,78,69,58)(52,63,77,70,57)(55,80,86,81,66)(56,79,85,82,65)(67,74,72,87,83)(68,73,71,88,84);(1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240917);