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×#13243 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C149 | #466 |
| commutator | C2×C8 | C22×C4×C149 |
| Frattini, cosocle | C22×C8 | C23×C149 |
| socle | C2×C149 | #1970 |
| prime radical | C2×#508×C149 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C149 ◁ C22×C4×C149 ◁ #405×C149 ◁ G | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 213 points:
(1,2)(3,15,20,39)(4,22)(5,12)(6,29)(7,31)(8,14)(9,18,34,27)(10,24)(11,16)(13,30)(17,32,47,58)(19,35,43,57)(21,33,49,42)(23,51)(25,36)(26,53)(28,40)(37,55)(38,44)(41,61,60,46)(45,62,54,50)(48,64,63,59)(52,56)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3,16,43,30,49,5,27,8,20,6,19,7,21,28,18)(2,9,12,33,31,57,11,39,14,34,40,42,13,35,29,15)(4,17,44,61,56,63,23,54,26,47,24,46,25,48,55,45)(10,32,22,50,51,64,36,60,38,58,53,62,37,59,52,41);(1,4,5,23,7,25,16,44,8,26,28,55,30,56,6,24)(2,10,29,52,13,37,40,53,14,38,11,36,31,51,12,22)(3,17,27,54,21,48,43,61,20,47,18,45,49,63,19,46)(9,32,15,41,35,59,42,62,34,58,39,60,57,64,33,50)7748 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6231);