Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C149 · Schur multiplier (H2) = C4 |
| direct product | C149×#13641 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C149 | C23 |
| commutator | C22 | C22×C16×C149 |
| Frattini, cosocle | C2×C16 | C23×C149 |
| socle | C22×C149 | C22×C16 |
| prime radical | C23×C149 | C2×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C149 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 221 |
| transitive degree | μt(G) = 19072 |
A minimal faithful permutation representation — 3 generators on 221 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,17,13,18,25,35,14,19,26,36,28,38,47,57,15,20,27,37,29,39,48,58,30,40,49,59,50,60,65,70)(11,16,22,32,23,33,44,54,24,34,45,55,46,56,64,69,21,31,41,51,42,52,61,66,43,53,62,67,63,68,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,32)(18,33)(19,34)(20,31)(25,44)(26,45)(27,41)(28,46)(29,42)(30,43)(35,54)(36,55)(37,51)(38,56)(39,52)(40,53)(47,64)(48,61)(49,62)(50,63)(57,69)(58,66)(59,67)(60,68)(65,71)(70,72);(1,3,4,7)(2,6,5,8)(11,21)(16,31)(22,41)(23,42)(24,43)(32,51)(33,52)(34,53)(44,61)(45,62)(46,63)(54,66)(55,67)(56,68)(64,71)(69,72)16688 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6629);