Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C109 · Schur multiplier (H2) = C24 |
| direct product | C2×C109×#1382 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C109 | D8 |
| commutator | C8 | C2×C42×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| socle | C23×C109 | #102 |
| prime radical | C24×C109 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C109 ◁ C23×C4×C109 ◁ C23×C8×C109 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 3 generators on 147 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(3,4,5,6)(7,8)(9,18)(10,23)(11,26)(12,28)(13,17)(14,19)(15,25)(16,24)(20,30)(21,38)(22,34)(27,31)(29,37)(32,36)(33,35);(7,9,11,20,24,35,25,36,13,22,10,19,12,21,27,37)(8,14,15,18,28,32,26,38,17,30,31,34,16,29,23,33)9592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5548);