Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C16 · Schur multiplier (H2) = C1 |
| direct product | C5×#68006 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C16 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C5×C16 |
| Frattini | C16 | C5×S4 |
| Fitting | C5×#504 | D3 |
| socle | C2×C5 | A4⋊C16 |
| prime radical | C5×#41⋊C3 | C8 |
| cosocle | #504⋊C3 | C2×C5 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C5×C16 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2,3,4,5)(6,35,62,41,28,44,106,103,10,77,108,85,67,89,131,128,13,17,64,61,9,71,104,82,31,47,110,107,25,114,129,118)(7,52,87,27,46,55,121,79,16,98,122,66,91,100,133,12,19,22,88,73,15,94,120,30,49,58,123,116,43,126,132,37)(8,63,69,45,57,81,78,74,21,109,112,90,102,117,33,18,24,40,72,68,20,105,75,48,60,84,115,111,54,130,36,32)(11,53,50,14,70,93,56,83,29,99,96,39,113,125,101,119,34,51,23,42,26,95,92,38,76,97,59,86,65,127,124,80);(11,12,36)(17,18,53)(22,23,64)(26,27,72)(29,30,78)(32,51,35)(33,34,37)(40,41,88)(44,45,95)(47,48,99)(50,62,52)(55,56,106)(58,59,110)(61,87,63)(65,66,115)(68,93,71)(69,70,73)(74,97,77)(75,76,79)(81,82,121)(84,85,123)(89,90,127)(92,104,94)(96,108,98)(100,101,131)(103,120,105)(107,122,109)(111,125,114)(112,113,116)(117,118,133)(124,129,126)(128,132,130)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236363);