Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C149 · Schur multiplier (H2) = C22 |
| direct product | C149×#7180 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C149 | #106 |
| commutator | C8 | C4×C8×C149 |
| Frattini, cosocle | C2×C4×C8 | C22×C149 |
| socle | C2×C149 | #1279 |
| prime radical | C23×C149 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C149 ◁ C22×C8×C149 ◁ C2×C4×C8×C149 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 4768 |
A minimal faithful permutation representation — 2 generators on 181 points:
(1,2,5,11,7,13,23,28,8,14,22,10,6,12,4,21)(3,15,17,26,19,31,16,27,20,29,24,25,18,30,32,9)(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)(2,9,12,25,14,27,13,26)(4,16)(5,24)(6,18)(7,19)(8,20)(10,15,28,30,11,29,21,31)(17,22)(23,32)9536 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 168);