Finite Group DatabasebetaFinite Group Databasebeta| order | 19072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C149 · Schur multiplier (H2) = C25 |
| direct product | #3044×C149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C149 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C149 |
| prime radical | C2×#41×C149 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C149 ◁ #117×C149 ◁ G | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 185 |
| transitive degree | μt(G) = 9536 |
A minimal faithful permutation representation — 4 generators on 185 points:
(2,4)(5,6,10,14)(7,11,16,25)(8,18,20,28)(9,21,23,13)(12,22,27,19)(15,29,30,36)(17,33,32,26)(24,34,35,31)(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)(6,11)(8,15)(9,17)(10,16)(12,24)(13,26)(14,25)(18,29)(19,31)(20,30)(21,33)(22,34)(23,32)(27,35)(28,36);(5,8,10,20)(6,12,14,27)(7,15,16,30)(9,22,23,19)(11,24,25,35)(13,28,21,18)(17,34,32,31)(26,36,33,29);(7,16)(11,25)(15,30)(17,32)(24,35)(26,33)(29,36)(31,34)4768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19072, 1776);