Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#73038 |
| Fitting | C5×#504 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C16 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C5×C8 |
| Frattini | C8 | C2×C5×S4 |
| Fitting | C5×#504 | D3 |
| socle | C2×C5 | C8×S4 |
| prime radical | C5×#985 | C4 |
| cosocle | #131⋊C3 | C22×C5 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C5×C16 | |
174 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,38,46,13,49,61,18,68,26,17,19,55,24,33,63,53,27,39,8,56,50,14,64,69,12,15,44,32,22,59,42,51,28,7,45,57,9,60,65,29,10,37,21,30,48,35,40,67)(11,16,20,31,23,34,41,52,25,36,43,54,47,58,62,66);(10,25)(11,27)(15,36)(16,38)(19,43)(20,45)(22,47)(23,49)(26,28)(30,54)(31,56)(33,58)(34,60)(37,39)(40,62)(41,64)(44,46)(48,50)(51,66)(52,68)(55,57)(59,61)(63,65)(67,69)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236460);