Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C149 · Schur multiplier (H2) = C25 |
| direct product | C149×#17409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C149 |
| Frattini, cosocle | C22×C4 | C24×C149 |
| prime radical | C2×#121×C149 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C2×C4×Q8×C149 ◁ G | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 4 generators on 181 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,22,25,23,26,21,30)(19,27,28,31,20,24,29,32)(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)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,24)(20,23)(21,28)(22,29)(25,31)(26,27)(30,32);(3,9)(6,12)(10,15)(13,16)(17,20,23,19)(18,24,26,27)(21,28,22,29)(25,32,30,31);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,25,26,30)(19,28,20,29)(24,31,27,32)6854 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 10397);