Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C24 |
| direct product | C149×#13017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | #447 |
| commutator | C2×C8 | C22×C4×C149 |
| Frattini, cosocle | C22×C8 | C23×C149 |
| prime radical | C22×D16×C149 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C22×C4×C149 ◁ #405×C149 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 185 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 185 points:
(2,4)(5,6,8,13,23,32,9,25,11,16,24,33,10,15,22,14)(7,17,18,34,27,29,36,31,21,35,26,30,20,28,19,12)(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);(1,2,3,4)(5,7)(6,12)(8,19)(9,26)(10,27)(11,21)(13,28)(14,17)(15,34)(16,31)(18,22)(20,23)(24,36)(25,35)(29,33)(30,32);(6,13)(8,22)(9,24)(10,23)(12,28)(14,32)(15,25)(16,33)(17,30)(18,19)(20,27)(26,36)(29,31)(34,35)6854 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6005);