Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12549 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C2×D4 |
| commutator | C4 | C2×C4×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×C4⋊C8 |
| prime radical | C24×C79 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C23×C8×C79 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(5,6,9,14,10,13,23,28,11,15,24,29,8,21,22,34)(7,16,19,31,20,26,33,35,17,30,32,36,18,12,25,27)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2,3,4)(5,7)(6,12,15,26)(8,18)(9,19)(10,20)(11,17)(13,16,21,30)(14,27,29,35)(22,25)(23,33)(24,32)(28,31,34,36);(7,17)(12,26)(16,30)(18,20)(19,32)(25,33)(27,35)(31,36)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5537);