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 | C4⋊C4 | S5 |
| prime radical | C4⋊#6732 | C2 |
| hypocenter | SL(2,𝔽5) | #41 |
| LCS | G ▷ C2×SL(2,𝔽5) ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
360 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 26 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:
(5,40,13,30,34,51,6,39,14,29,33,52)(7,37,15,32,36,50,8,38,16,31,35,49)(9,47,23,18,27,41,10,48,24,17,28,42)(11,46,22,20,26,43,12,45,21,19,25,44)(53,54)(55,56)(57,91,65,81,86,104,58,92,66,82,85,103)(59,90,67,83,88,101,60,89,68,84,87,102)(61,100,75,69,79,94,62,99,76,70,80,93)(63,97,74,71,78,96,64,98,73,72,77,95);(1,3,2,4)(5,31,36,44,28,51,6,32,35,43,27,52)(7,30,33,41,25,50,8,29,34,42,26,49)(9,47,16,40,22,19,10,48,15,39,21,20)(11,46,13,37,24,18,12,45,14,38,23,17)(53,55,54,56)(57,83,88,96,80,103,58,84,87,95,79,104)(59,82,85,93,77,102,60,81,86,94,78,101)(61,99,68,92,74,71,62,100,67,91,73,72)(63,98,65,89,76,70,64,97,66,90,75,69);(1,53)(2,54)(3,55)(4,56)(5,57,6,58)(7,59,8,60)(9,61,10,62)(11,63,12,64)(13,65,14,66)(15,67,16,68)(17,69,18,70)(19,71,20,72)(21,73,22,74)(23,75,24,76)(25,77,26,78)(27,79,28,80)(29,81,30,82)(31,83,32,84)(33,85,34,86)(35,87,36,88)(37,89,38,90)(39,91,40,92)(41,93,42,94)(43,95,44,96)(45,97,46,98)(47,99,48,100)(49,101,50,102)(51,103,52,104)48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240941);