Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#7872 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C79 | D4 |
| commutator | C4 | C2×C4×C8×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| socle | C23×C79 | C2×C4⋊C4 |
| prime radical | C24×C79 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C79 ◁ C22×C42×C79 ◁ G | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 107 points:
(5,6,7,9,8,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,2,3,4)(13,15,16,24,18,26,17,25)(14,19,21,27,22,28,20,23);(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,23,28,27)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 860);