Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C24×C4 |
| direct product | C79×#17404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C24×C79 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C42×C79 ◁ G | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2)(3,11,14,7)(4,8)(5,9)(6,10)(12,16,13,15)(17,18,20,24,21,25,28,23)(19,22,27,31,26,29,32,30)(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);(1,3,6,14)(2,7,10,11)(4,12,5,13)(8,15,9,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,31)(25,29)(28,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(19,26)(22,29)(27,32)(30,31);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(17,20,21,28)(18,23,25,24)(19,27,26,32)(22,30,29,31)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 10392);