Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C22×C4 |
| direct product | C109×#9310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | #386 |
| commutator | C42 | C22×C4×C109 |
| Frattini, cosocle | C2×C42 | C23×C109 |
| prime radical | #532×C109 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 149 points:
(2,6)(3,4)(7,8)(9,10)(11,15,20,31)(12,22)(13,26)(14,17)(16,24)(18,28)(19,35,23,40)(21,38,37,30)(25,34)(27,33)(29,39,32,36)(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);(1,2,3,8,5,7,4,6)(9,11,14,20)(10,15)(12,23,25,19)(13,21,28,37)(16,32)(17,31)(18,30)(22,40)(24,39,27,36)(26,38)(29,33)(34,35);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,17,33)(11,19,20,23)(13,27,28,24)(15,29,31,32)(18,34,26,22)(21,39,37,36)(30,40,38,35)4033 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 2298);