Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12958 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C79 |
| Frattini, cosocle | C22×D4 | C23×C79 |
| prime radical | C2×#120×C79 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C79 ◁ C22×D4×C79 ◁ G | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 91 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(1,3,4,2)(6,9)(10,12);(7,11)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5946);