Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C2×C4 |
| direct product | C149×#11866 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | C2×#102 |
| commutator | C2×C8 | C22×C4×C149 |
| Frattini, cosocle | C22×C8 | C23×C149 |
| prime radical | #466×C149 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C2×C42×C149 ◁ C2×C4×C8×C149 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 197 |
| transitive degree | μt(G) = 19072 |
A minimal faithful permutation representation — 3 generators on 197 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18,22,28,24,30,23,29)(19,31,34,46,36,47,35,42)(20,37,21,40,39,27,38,26)(25,41,43,33,45,32,44,48)(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);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,25)(20,33)(21,41)(22,35)(23,34)(24,36)(26,31)(27,46)(28,44)(29,43)(30,45)(32,38)(37,42)(39,48)(40,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,23,38,24,39,22,21)(18,26,28,37,30,40,29,27)(19,32,35,33,36,41,34,48)(25,42,43,31,45,46,44,47)5960 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 4854);