Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S5 |
| commutator, cosocle | C2×SL(2,𝔽5) | C23 |
| Fitting, hypercenter | C2×Q8 | S5 |
| hypocenter | SL(2,𝔽5) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽5) ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
554 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 554 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) = 160 |
Permutation generators:
(1,3,2,4)(5,42,35,14,40,34)(6,41,36,13,39,33)(7,16,22,8,15,21)(9,17,29)(10,18,30)(11,28,24,25,44,20)(12,27,23,26,43,19)(31,37)(32,38)(45,47,46,48)(49,86,79,58,84,78)(50,85,80,57,83,77)(51,60,66,52,59,65)(53,61,73)(54,62,74)(55,72,68,69,88,64)(56,71,67,70,87,63)(75,81)(76,82);(1,4)(2,3)(5,29,11,15,6,30,12,16)(7,44,23,10,8,43,24,9)(13,28,38,20,14,27,37,19)(17,40,35,21,18,39,36,22)(25,41,32,33,26,42,31,34)(45,48)(46,47)(49,74,55,60,50,73,56,59)(51,87,67,53,52,88,68,54)(57,71,82,63,58,72,81,64)(61,83,79,66,62,84,80,65)(69,86,76,78,70,85,75,77);(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, 240800);