Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C149 · Schur multiplier (H2) = C26 |
| direct product | C149×#10753 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C149 | #120 |
| commutator | C2×C4 | C2×C42×C149 |
| Frattini, cosocle | C23×C4 | C23×C149 |
| socle | C22×C149 | #385 |
| prime radical | C22×#41×C149 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C149 ◁ C23×C4×C149 ◁ G | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 185 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 185 points:
(1,2,3,4)(5,6,9,15,11,16,10,14)(7,17,21,26,20,27,19,12)(8,13,23,28,24,30,22,29)(18,32,33,25,34,31,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(5,7,10,19,11,20,9,21)(6,12,14,27,16,26,15,17)(8,18,22,35,24,34,23,33)(13,25,29,32,30,36,28,31);(5,8)(6,13)(7,18)(9,23)(10,22)(11,24)(12,25)(14,29)(15,28)(16,30)(17,31)(19,35)(20,34)(21,33)(26,36)(27,32)9536 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 3741);