Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#32699 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24×C109 |
| prime radical | C2×AGL(1,ℤ/8)×C109 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C22×C4×C109 ◁ G | |
538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2,7,12,6,11,19,27)(3,4,9,16,14,8,20,23)(5,10,18,26,17,25,21,28)(13,22,29,32,30,15,24,31)(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)(4,11)(5,13)(6,14)(7,20)(9,19)(10,15)(12,16)(17,30)(18,24)(21,29)(22,25)(23,27)(26,32)(28,31);(1,4)(2,9,11,20)(3,12,14,27)(5,15)(6,8)(7,16)(10,24,25,29)(13,28,30,26)(17,22)(18,31)(19,23)(21,32);(1,5,6,17)(2,10,11,25)(3,13,14,30)(4,15,8,22)(7,21,19,18)(9,24,20,29)(12,28,27,26)(16,32,23,31)3706 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 25687);