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 | |
418 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 28 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:
(1,4,2,3)(5,23,30,9,28,7,22,32,11,25,6,24,29,10,27,8,21,31,12,26)(13,20,37,44,36,15,17,39,41,33,14,19,38,43,35,16,18,40,42,34)(45,47,46,48)(49,67,74,53,72,51,66,76,55,69,50,68,73,54,71,52,65,75,56,70)(57,64,81,88,80,59,61,83,85,77,58,63,82,87,79,60,62,84,86,78);(1,4,2,3)(5,33,12,26,6,34,11,25)(7,36,9,27,8,35,10,28)(13,42,24,31,14,41,23,32)(15,43,21,29,16,44,22,30)(17,40,20,37,18,39,19,38)(45,48,46,47)(49,77,56,70,50,78,55,69)(51,80,53,71,52,79,54,72)(57,86,68,75,58,85,67,76)(59,87,65,73,60,88,66,74)(61,84,64,81,62,83,63,82);(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)48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240936);