Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1790 |
| commutator | C2×Q8 | C22×C4×C109 |
| Frattini, cosocle | C22×Q8 | C23×C109 |
| prime radical | C109×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C23×C109 ◁ C22×Q8×C109 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,7,13,8,14,6,12)(3,15,17,31,18,29,23,25)(4,19,21,10,5,11,20,28)(9,24,26,16,27,32,30,22)(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)(2,9)(4,16)(5,22)(6,23)(7,17)(8,18)(10,25)(11,29)(12,26)(13,30)(14,27)(15,19)(20,24)(21,32)(28,31);(1,4,8,5)(2,10,14,28)(3,16,18,22)(6,21,7,20)(9,25,27,31)(11,13,19,12)(15,26,29,30)(17,24,23,32)3706 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5770);