Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C4 |
| direct product | C79×#7729 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C79 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C43×C79 ◁ G | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,20,24,21,23,29,31)(19,25,27,32,26,22,28,30)(33,34)(35,41,43,48)(36,40)(37,39)(38,42,46,44)(45,47)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,22)(20,28)(21,26)(23,25)(24,32)(27,29)(30,31)(33,35,37,43)(34,38,39,46)(36,44,45,42)(40,48,47,41);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,23)(19,26)(24,31)(27,28)(34,39)(38,46)(40,47)(41,48)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 717);