Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#17683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C109 |
| Frattini, cosocle | C42 | C24×C109 |
| prime radical | C109×#3518 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C4×#41×C109 ◁ G | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,26,23,19,21,25)(20,27,28,32,29,24,30,31)(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)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,32)(30,31);(2,7)(6,14)(8,15)(9,16)(17,20,23,29)(18,24,19,27)(21,30,22,28)(25,32,26,31);(17,21,23,22)(18,25,19,26)(20,28,29,30)(24,31,27,32)5341 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 10671);