Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C24 |
| direct product | C149×#13103 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | #1898 |
| commutator | C22×C4 | C22×C4×C149 |
| Frattini, cosocle | C2×#41 | C23×C149 |
| prime radical | #3010×C149 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C2×C4×C149 ◁ C2×2−1+4×C149 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 4768 |
A minimal faithful permutation representation — 3 generators on 181 points:
(1,2,7,12,8,13,6,11)(3,5,10,23,16,24,25,14)(4,17,20,9,21,27,19,28)(15,30,26,18,31,32,29,22)(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);(1,3,14,2)(4,15,30,9)(5,12,6,25)(7,10,24,11)(8,16,23,13)(17,20,26,22)(18,27,19,29)(21,31,32,28);(1,4,8,21)(2,9,13,28)(3,15,16,31)(5,22,24,18)(6,20,7,19)(10,29,25,26)(11,27,12,17)(14,30,23,32)5066 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6091);