Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C149 · Schur multiplier (H2) = C25 |
| direct product | C149×#17422 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C149 |
| Frattini, cosocle | C22×C4 | C24×C149 |
| prime radical | C2×#120×C149 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C2×C4×D4×C149 ◁ G | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 4768 |
A minimal faithful permutation representation — 4 generators on 181 points:
(1,2,4,9,6,10,16,24)(3,8,12,22,14,23,27,32)(5,7,15,19,17,20,28,30)(11,21,25,31,13,18,26,29)(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);(2,7)(4,15)(8,18)(10,20)(12,26)(16,28)(21,23)(25,27);(1,3,5,13)(2,8,7,18)(4,12,15,26)(6,14,17,11)(9,22,19,29)(10,23,20,21)(16,27,28,25)(24,32,30,31);(3,11)(8,21)(12,25)(13,14)(18,23)(22,31)(26,27)(29,32)8195 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 10410);