Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C25 |
| direct product | C149×#11094 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C149 | #392 |
| commutator | C22×C4 | C22×C4×C149 |
| Frattini, cosocle | C23×C4 | C23×C149 |
| socle | C2×C149 | #1439 |
| prime radical | C22×#41×C149 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C149 ◁ C23×C4×C149 ◁ G | |
868 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)(3,8,15,23,14,24,13,29)(4,16,19,9,18,26,21,25)(12,27,28,22,17,31,30,32)(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)(4,17,18,12)(5,13)(6,15)(7,14)(9,22,25,32)(10,23)(11,24)(16,27,26,31)(19,30,21,28)(20,29);(1,4)(2,9)(3,12)(5,21)(6,19)(7,18)(8,22)(10,26)(11,25)(13,30)(14,17)(15,28)(16,20)(23,31)(24,32)(27,29)6854 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 4082);