Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#11158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | #397 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C23×C4 | C23×C109 |
| prime radical | C24×C109 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C23×C4×C109 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 145 points:
(1,2)(3,4)(5,6)(7,16)(8,21,25,36)(9,26)(10,14)(11,15)(12,18)(13,24,29,23)(17,32,22,27)(19,28)(20,30)(31,34,35,33)(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);(5,7,11,20)(6,12,15,28)(8,22,25,17)(9,19,10,18)(13,27,29,32)(14,30,26,16)(21,35,36,31)(23,33,24,34);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,21,15,29,26,36)(7,17,18,33,20,22,19,34)(12,27,16,31,28,32,30,35)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4146);