Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C149 · Schur multiplier (H2) = C22×C4 |
| direct product | C149×#13242 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C149 | C2×D8 |
| commutator | C2×C8 | C22×C4×C149 |
| Frattini, cosocle | C22×C8 | C23×C149 |
| socle | C22×C149 | C4×D8 |
| prime radical | C2×#508×C149 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C149 ◁ C22×C8×C149 ◁ C2×C4×C8×C149 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 185 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 185 points:
(2,4)(5,6,10,15,11,17,27,16)(7,13,20,30,21,32,36,31)(8,22,24,29,25,28,19,12)(9,26,23,14,18,33,35,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185);(1,2,3,4)(5,7,18,24,27,36,23,8,11,21,9,19,10,20,35,25)(6,12,14,13,15,22,33,30,17,29,34,32,16,28,26,31);(5,8,23,20,10,24,18,21,11,25,35,36,27,19,9,7)(6,13,26,12,16,31,34,28,17,32,33,29,15,30,14,22)7748 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 6230);