Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#12012 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C79 | D4 |
| commutator | C4 | C22×C16×C79 |
| Frattini, cosocle | C4×C8 | C23×C79 |
| socle | C22×C79 | C2×C4⋊C8 |
| prime radical | C23×C79 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18)(19,26)(20,23)(21,25)(22,27)(24,29)(28,30)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(17,19,20,27,21,28,29,32)(18,22,24,26,25,31,23,30);(17,20,21,29)(18,23,25,24)(19,27,28,32)(22,30,31,26)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5000);