Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C23 |
| direct product | C2×#1420×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | #357 |
| commutator | C16 | C22×C4×C149 |
| Frattini, cosocle | C2×C16 | C23×C149 |
| prime radical | C23×C149 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ C22×C4×C149 ◁ C22×C8×C149 ◁ C22×C16×C149 ◁ G | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 215 |
| transitive degree | μt(G) = 19072 |
A minimal faithful permutation representation — 3 generators on 215 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215);(3,4,7,13,10,15,9,33)(5,16,18,47,21,49,20,56)(6,22,25,57,28,59,27,64)(8,29,24,45,32,40,31,14)(11,34,35,17,37,51,43,53)(12,38,41,63,42,62,58,26)(19,54,50,39,55,66,30,48)(23,44,60,65,52,46,61,36);(3,5,10,21)(4,11,15,37)(6,23,28,52)(7,20,9,18)(8,30,32,50)(12,39,42,48)(13,43,33,35)(14,44,45,46)(16,22,49,59)(17,26,53,63)(19,24,55,31)(25,61,27,60)(29,36,40,65)(34,38,51,62)(41,54,58,66)(47,64,56,57)7748 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6612);