Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#13002 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | C2×#102 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| prime radical | C109×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C23×C4×C109 ◁ C23×C8×C109 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 145 points:
(1,2)(3,4)(5,6,10,15,11,16,23,33,12,17,9,25,8,14,24,31)(7,18,21,29,19,32,27,30,22,35,20,34,28,36,26,13)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(5,7)(6,13,17,30)(8,19)(9,26)(10,27)(11,28)(12,22)(14,29,16,34)(15,32,25,36)(18,31,35,33)(20,24)(21,23);(2,4)(5,8,12,11)(6,14,17,16)(7,19,22,28)(9,23,10,24)(13,29,30,34)(15,31,25,33)(18,32,35,36)(20,26,21,27)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5990);