Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C109 · Schur multiplier (H2) = C24×C4 |
| direct product | C109×#17609 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C109 |
| Frattini, cosocle | C22×C4 | C24×C109 |
| prime radical | #525×C109 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C2×#117×C109 ◁ G | |
944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 4 generators on 141 points:
(2,7,9,21)(4,15)(8,20,24,31)(10,22,11,16)(13,28)(17,19)(23,29,25,32)(26,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,2,5,10)(3,8,14,25)(4,16,15,21)(6,11,18,9)(7,19,22,17)(12,23,27,24)(13,29,28,31)(20,30,32,26);(1,3,5,14)(2,8,10,25)(4,13,15,28)(6,12,18,27)(7,20,22,32)(9,24,11,23)(16,29,21,31)(17,26,19,30);(3,12)(8,23)(13,26)(14,27)(20,29)(24,25)(28,30)(31,32)5668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 10597);