Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12741 |
| 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 | |
930 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)(3,12,14,8)(4,7)(5,19)(6,10)(9,16)(11,17)(13,25,30,20)(15,24,29,27)(18,21)(22,31)(23,28,26,32)(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);(2,7)(4,16)(5,17)(8,20)(9,11)(10,21)(12,25)(13,28)(15,29)(18,31)(19,22)(23,24)(26,27)(30,32);(1,3)(2,8)(4,13)(5,15)(6,14)(7,20)(9,23)(10,12)(11,24)(16,28)(17,29)(18,30)(19,27)(21,25)(22,26)(31,32)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5729);