Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C149 · Schur multiplier (H2) = C23×C4 |
| direct product | C149×#10803 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C149 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C149 |
| Frattini, cosocle | C23×C4 | C23×C149 |
| socle | C22×C149 | C4×C22⋊C4 |
| prime radical | C24×C149 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C149 ◁ C23×C8×C149 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 185 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 3 generators on 185 points:
(2,3)(5,6)(7,11)(8,18,19,12)(9,14)(10,13)(15,27,28,22)(16,24)(17,23)(20,31,32,25)(21,26)(29,35,36,33)(30,34)(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);(1,2,4,3)(5,7,9,16,10,17,21,30)(6,11,14,24,13,23,26,34)(8,15,20,29,19,28,32,36)(12,22,25,33,18,27,31,35);(1,3,4,2)(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,18)(14,25)(16,29)(17,28)(21,32)(23,27)(24,33)(26,31)(30,36)(34,35)9536 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 3791);