Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33373 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C22×#99 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C2×Q8 | C24×C79 |
| prime radical | C79×#3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×Q8×C79 ◁ C2×H(5,𝔽2)×C79 ◁ G | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 95 |
| transitive degree | μt(G) = 1264 |
A minimal faithful permutation representation — 4 generators on 95 points:
(1,2,3,8,5,9,12,14)(4,7,10,6,13,15,11,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(1,3,5,12)(2,6,9,16)(4,10,13,11)(7,14,15,8);(1,4,5,13)(2,7,9,15)(3,10,12,11)(6,14,16,8);(1,3,5,12)(2,8,9,14)(4,11,13,10)(6,7,16,15)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26361);