Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#9418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #386 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | #532×C79 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2)(3,10,13,16)(4,14)(5,8)(6,9)(7,12,15,11)(17,18,22,26)(19,20)(21,25,27,32)(23,31)(24,30)(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);(1,2,5,14,6,9,4,8)(3,7,11,10,13,15,12,16)(17,19,27,28,22,24,21,31)(18,23,32,20,26,29,25,30);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,20)(18,24)(19,26)(21,23)(22,30)(25,28)(27,29)(31,32)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 2406);