Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C149 · Schur multiplier (H2) = C2 |
| direct product | C149×#7332 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C149 | C42 |
| commutator | C4 | C4×C16×C149 |
| Frattini, cosocle | C4×C16 | C22×C149 |
| socle, prime radical | C22×C149 | #346 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C149 ◁ G | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 189 |
| transitive degree | μt(G) = 38144 |
A minimal faithful permutation representation — 2 generators on 189 points:
(1,2,3,6,4,5,7,8)(9,10,12,19,14,20,15,13,17,22,32,33,16,21,31,34)(11,23,24,39,26,37,40,38,29,36,30,25,28,18,27,35)(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);(2,5)(6,8)(9,11,13,25,15,27,21,36,16,28,19,38,12,30,22,37,17,29,34,39,31,40,20,23,14,26,33,35,32,24,10,18)13112 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 320);