Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C22×C4 |
| direct product | C109×#12907 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C109 | C2×D8 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C2×C42×C109 ◁ C2×C4×C8×C109 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 27904 |
A minimal faithful permutation representation — 3 generators on 149 points:
(2,4)(5,6,7,8)(9,10)(11,19)(12,24)(13,16)(14,27)(15,21)(17,20)(18,25)(22,31)(23,38)(26,39)(28,37)(29,40)(30,35)(32,34)(33,36)(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);(9,11,20,34,14,23,26,37,13,22,35,40,25,36,12,21)(10,15,24,33,18,29,30,31,16,28,39,38,27,32,17,19);(1,2,3,4)(10,16)(15,28)(17,30)(18,27)(19,31)(24,39)(29,32)(33,38)8284 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5895);