Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#11272 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | #121 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C23×C4 | C23×C109 |
| socle | C22×C109 | #389 |
| prime radical | C24×C109 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C23×C4×C109 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 145 points:
(1,2,4,3)(5,6,10,14,11,16,9,15)(7,17,20,28,21,31,19,12)(8,22,25,30,26,32,24,13)(18,34,35,29,36,27,23,33)(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);(1,3,4,2)(5,7,9,19,11,21,10,20)(6,12,14,17,16,28,15,31)(8,23,24,36,26,35,25,18)(13,29,22,27,30,33,32,34);(2,3)(5,8,9,24,11,26,10,25)(6,13,14,22,16,30,15,32)(7,18,19,23,21,36,20,35)(12,27,17,33,28,34,31,29)5668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4260);