Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C109 · Schur multiplier (H2) = C2 |
| direct product | C109×#7461 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | D16 |
| commutator | C16 | C2×C8×C109 |
| Frattini, cosocle | C4×C16 | C22×C109 |
| socle, prime radical | C22×C109 | #366 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C42×C109 ◁ C4×C8×C109 ◁ C4×C16×C109 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 27904 |
A minimal faithful permutation representation — 2 generators on 149 points:
(1,2,3,5,4,6,7,8)(10,15)(11,20)(12,21)(13,24)(16,23)(17,32)(18,35)(19,29)(22,31)(25,38)(26,27)(28,33)(30,39)(34,40)(36,37)(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);(9,10,16,30,12,17,22,28,24,33,37,32,38,39,11,15,14,19,20,27,25,34,36,35,13,18,31,40,21,26,23,29)8284 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 449);