Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C24 |
| direct product | C149×#11284 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C149 | #397 |
| commutator | C22×C4 | C22×C4×C149 |
| Frattini, cosocle | C23×C4 | C23×C149 |
| socle | C2×C149 | #1442 |
| prime radical | C24×C149 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C149 ◁ C23×C4×C149 ◁ G | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 4768 |
A minimal faithful permutation representation — 3 generators on 181 points:
(1,2,6,10,7,11,5,20)(3,12,14,8,15,24,21,23)(4,16,19,31,18,9,17,25)(13,27,28,30,29,26,32,22)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181);(1,3)(2,8)(4,13,18,29)(5,21)(6,14)(7,15)(9,22,16,30)(10,24)(11,23)(12,20)(17,32,19,28)(25,27,31,26);(1,4,5,17,7,18,6,19)(2,9,10,25,11,16,20,31)(3,13,14,28,15,29,21,32)(8,22,12,26,23,30,24,27)6854 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 4272);