Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#19277 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C109 |
| prime radical | C2×#41×C109 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 4 generators on 141 points:
(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(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);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(18,23)(24,31)(25,26)(27,32);(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)5668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 12265);