Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1272 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C79 | C22⋊C4 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C79 ◁ C22×C42×C79 ◁ G | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(3,4,5,7,6,8,9,10)(11,12)(13,17,18,15)(14,16)(20,23,24,25);(11,13)(12,15)(14,18)(16,17)(19,20,22,24)(21,25,26,23)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 549);