Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33571 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C79 |
| prime radical | C79×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×#41×C79 ◁ G | |
1084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2)(3,13)(4,9)(5,8)(6,12)(7,14)(10,19)(11,21)(15,29)(16,24)(17,26)(18,27)(20,30)(22,28)(23,31)(25,32)(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);(1,3,6,16)(2,7,12,26)(4,17,19,14)(5,20,21,15)(8,23,11,25)(9,13,10,24)(18,31,22,32)(27,29,28,30);(2,8)(5,21)(7,23)(9,27)(10,28)(11,12)(13,29)(15,20)(18,22)(24,30)(25,26)(31,32);(1,4,6,19)(2,9,12,10)(3,14,16,17)(5,18,21,22)(7,24,26,13)(8,27,11,28)(15,31,20,32)(23,30,25,29)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26559);