Finite Group DatabasebetaFinite Group Databasebeta| order | 22144 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C173 · Schur multiplier (H2) = C1 |
| direct product | #1356×C173 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C173 | #329 |
| commutator | C42 | C2×C4×C173 |
| Frattini, cosocle | #97 | C22×C173 |
| prime radical | C22×C173 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C173 ◁ C2×C4×C173 ◁ #97×C173 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 205 |
| transitive degree | μt(G) = 5536 |
A minimal faithful permutation representation — 2 generators on 205 points:
(1,2,4,9,5,10,18,28,6,11,17,29,20,8,25,26)(3,12,13,30,14,7,21,22,15,23,31,27,19,24,16,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205);(1,3,6,15)(2,7,11,24)(4,16,17,21)(5,19,20,14)(8,23,10,12)(9,27,29,30)(13,18,31,25)(22,26,32,28)3460 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22144, 88);