Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#11305 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | #124 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C23×C4 | C23×C109 |
| socle | C22×C109 | #397 |
| prime radical | C24×C109 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C23×C4×C109 ◁ G | |
546 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)(8,13)(9,24)(10,15)(11,14)(12,18)(17,32)(19,33)(20,27)(21,26)(22,30)(23,31)(25,29)(28,34)(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);(5,7,11,20,10,19,9,18)(6,12,24,33,15,27,14,16)(8,21,22,34,23,35,25,17)(13,28,29,26,31,32,30,36);(5,8,9,22,10,23,11,25)(6,13,14,29,15,31,24,30)(7,17,18,21,19,34,20,35)(12,26,16,32,27,36,33,28)5668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4293);