Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | #121×C79 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(2,6)(4,11)(7,13)(8,12)(9,15)(14,16)(17,18,22,21)(19,26,30,32)(20,24,31,25)(23,27,28,29)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(4,11)(6,12)(9,15)(13,16)(17,19,22,30)(18,23,21,28)(20,27,31,29)(24,26,25,32);(1,2)(3,7)(4,12)(5,8)(6,11)(9,16)(10,14)(13,15)(17,20,22,31)(18,24,21,25)(19,27,30,29)(23,26,28,32);(1,3,5,10)(2,7,8,14)(4,9,11,15)(6,13,12,16)(17,21)(18,22)(19,28)(20,24)(23,30)(25,31)(26,27)(29,32)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11733);