Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#13018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | #447 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| prime radical | C22×D8×C109 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C22×C4×C109 ◁ #405×C109 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 145 points:
(2,4)(5,6,11,18,14,23,10,22,13,19,9,16,12,7,20,17)(8,24,27,15,31,33,26,21,29,36,25,35,28,32,30,34)(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);(1,2,3,4)(5,7,13,23)(6,14,19,12)(8,21,29,34)(9,18,11,16)(10,22,20,17)(15,28,35,31)(24,30,36,26)(25,32,27,33);(5,8,13,29)(6,15,19,35)(7,21,23,34)(9,26,11,30)(10,25,20,27)(12,31,14,28)(16,36,18,24)(17,33,22,32)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 6006);