Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C2×C4 |
| direct product | C149×#8603 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C149 | #117 |
| commutator | C22×C4 | C22×C4×C149 |
| Frattini, cosocle | C2×C42 | C23×C149 |
| socle | C22×C149 | #380 |
| prime radical | C24×C149 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C149 ◁ C2×#97×C149 ◁ G | |
368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 197 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 197 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,35)(28,41,43,39)(30,40)(32,38,47,46)(36,42,45,48)(37,44)(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)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,29,22,30,35,44)(18,23,25,27,26,37,34,40)(20,28,32,42,33,43,47,48)(24,36,38,39,31,45,46,41);(3,9)(6,12)(10,15)(13,16)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,38)(26,31)(27,39)(29,42)(30,43)(34,46)(35,47)(37,45)(40,41)(44,48)7748 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 1591);