Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#9312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | #386 |
| commutator | C42 | C22×C4×C109 |
| Frattini, cosocle | C2×C42 | C23×C109 |
| prime radical | C2×#41×C109 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 3 generators on 149 points:
(1,2,3,5,4,7,8,6)(9,10)(11,20,23,31)(12,24)(13,18)(14,28)(15,19)(16,22,32,27)(17,25)(21,36,39,30)(26,33)(29,34)(35,37,40,38)(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);(2,5,7,6)(9,11,13,27,15,23,14,22)(10,16,18,31,19,32,28,20)(12,21,25,38,26,39,29,37)(17,30,33,40,34,36,24,35);(9,12)(10,17)(11,21)(13,25)(14,29)(15,26)(16,30)(18,33)(19,34)(20,35)(22,37)(23,39)(24,28)(27,38)(31,40)(32,36)5341 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 2300);