Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12743 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1786 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| prime radical | C2×#576×C109 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C22×C4×C109 ◁ C2×#131×C109 ◁ G | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,5,12,7,11,6,23)(3,13,15,29,16,9,22,28)(4,17,19,30,21,24,20,10)(8,18,26,31,25,32,27,14)(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);(1,3)(2,8)(4,18)(5,15)(6,22)(7,16)(9,24)(10,28)(11,25)(12,26)(13,17)(14,20)(19,31)(21,32)(23,27)(29,30);(1,3)(2,9)(4,14)(5,22)(6,15)(7,16)(8,24)(10,27)(11,13)(12,29)(17,25)(18,20)(19,32)(21,31)(23,28)(26,30)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5731);