Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C149 · Schur multiplier (H2) = C1 |
| direct product | C149×#7457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C149 | #1368 |
| commutator | C16 | C2×C8×C149 |
| Frattini, cosocle | #346 | C22×C149 |
| prime radical | C22×C149 | #365 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C149 ◁ C2×C4×C149 ◁ C4×C8×C149 ◁ #346×C149 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 4768 |
A minimal faithful permutation representation — 2 generators on 181 points:
(2,7,8,17,10,19,22,20)(3,11,12,25)(4,14)(5,16)(9,23,21,18,24,31,30,32)(13,28,27,26)(15,29)(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);(1,2,3,7,4,9,11,18,5,8,13,20,14,21,26,23,6,10,12,19,15,24,25,32,16,22,27,17,29,30,28,31)5066 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 445);