Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C22×C4 |
| direct product | C149×#13236 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C149 | C2×D8 |
| commutator | C2×C8 | C22×C4×C149 |
| Frattini, cosocle | C22×C8 | C23×C149 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C149 ◁ C2×C42×C149 ◁ C2×C4×C8×C149 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 189 |
| transitive degree | μt(G) = 38144 |
A minimal faithful permutation representation — 3 generators on 189 points:
(2,4)(5,6,7,8)(9,10)(11,22)(12,28)(13,31)(14,32)(15,21)(16,24)(17,25)(18,30)(19,23)(20,29)(26,40)(27,33)(34,37)(35,39)(36,38)(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);(1,2,3,4)(6,8)(9,11,23,37,14,26,12,24,15,27,30,39,29,38,13,25)(10,16,31,40,20,34,18,22,21,17,28,36,32,35,19,33);(9,11,23,37,14,26,12,24,15,27,30,39,29,38,13,25)(10,17,31,36,20,35,18,33,21,16,28,40,32,34,19,22)11324 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6224);