Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#11591 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C109 | D8 |
| commutator | C8 | C22×C8×C109 |
| Frattini, cosocle | C4×C8 | C23×C109 |
| socle | C22×C109 | C2×#102 |
| prime radical | C2×#135×C109 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C109 ◁ C2×C42×C109 ◁ C2×C4×C8×C109 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 149 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 3 generators on 149 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20)(12,17)(13,25)(14,29)(15,19)(16,22)(18,27)(21,36)(23,37)(24,35)(26,31)(28,39)(30,34)(32,40)(33,38)(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);(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,34)(19,33)(20,25)(22,27)(24,38)(28,40)(29,32)(35,36)(37,39);(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,31,33,32)(18,25,35,39)(20,36,37,34)(22,26,38,40)9592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4579);