Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C149 · Schur multiplier (H2) = C2×C4 |
| direct product | C149×#13123 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C149 | #121 |
| commutator | C2×C4 | C22×C8×C149 |
| Frattini, cosocle | C22×C8 | C23×C149 |
| socle | C2×C149 | #1923 |
| prime radical | C2×#41×C149 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C149 ◁ C22×C8×C149 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 213 points:
(1,2,6,12,7,11,27,30,8,13,28,37,5,26,15,41)(3,14,18,39,19,33,48,58,20,40,49,61,17,9,29,32)(4,21,24,52,25,51,53,62,23,10,34,36,22,35,42,55)(16,43,46,63,47,31,56,57,45,38,59,60,44,50,54,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,3,15,29,5,17,28,49,8,20,27,48,7,19,6,18)(2,9,30,39,26,40,12,32,13,33,41,61,11,14,37,58)(4,16,42,54,22,44,34,59,23,45,53,56,25,47,24,46)(10,31,55,60,51,43,36,57,21,50,62,63,35,38,52,64);(1,4,5,22,8,23,7,25)(2,10,11,35,13,21,26,51)(3,16,17,44,20,45,19,47)(6,24,15,42,28,34,27,53)(9,31,14,38,33,50,40,43)(12,36,30,55,37,52,41,62)(18,46,29,54,49,59,48,56)(32,57,39,60,58,64,61,63)9536 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6111);