Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C109 · Schur multiplier (H2) = C22×C4 |
| direct product | C109×#8157 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C109 |
| Frattini, cosocle | C2×C42 | C23×C109 |
| socle | C22×C109 | C4×C22⋊C4 |
| prime radical | C24×C109 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C2×C4×C8×C109 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 149 points:
(9,10,13,17,14,18,25,31)(11,19,22,33,21,15,26,28)(12,16,23,29,24,30,36,39)(20,32,35,40,34,27,37,38)(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);(1,2,3,6,4,5,7,8)(9,11)(10,15,18,19)(12,20)(13,22)(14,21)(16,27,30,32)(17,28,31,33)(23,35)(24,34)(25,26)(29,38,39,40)(36,37);(2,5)(6,8)(9,12,13,23,14,24,25,36)(10,16,17,29,18,30,31,39)(11,20,22,35,21,34,26,37)(15,27,28,38,19,32,33,40)6976 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 1145);