Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C79 · Schur multiplier (H2) = C2 |
| direct product | C79×#7546 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C79 | D32 |
| commutator | C32 | C2×C4×C79 |
| Frattini, cosocle | C2×C32 | C22×C79 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C4×C79 ◁ C2×C8×C79 ◁ C2×C16×C79 ◁ C2×C32×C79 ◁ G | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 2 generators on 147 points:
(1,2,3,4)(6,12)(7,18)(8,19)(9,23)(10,26)(13,22)(14,37)(15,41)(16,44)(17,32)(20,51)(21,36)(24,55)(25,50)(27,54)(28,29)(30,38)(31,42)(33,58)(34,35)(39,67)(40,62)(43,61)(45,46)(47,52)(48,49)(53,64)(56,65)(57,63)(59,68)(60,66)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(5,6,13,33,8,14,34,61,23,38,64,67,49,57,52,44,10,16,36,63,24,39,20,30,27,43,46,37,50,58,7,12,11,17,18,29,25,40,45,41,54,66,51,65,55,68,21,31,26,42,47,59,48,56,53,60,9,15,35,62,19,28,22,32)5530 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 534);