Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#11246 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #391 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C23×C4 | C23×C109 |
| socle | C2×C109 | #1441 |
| prime radical | C24×C109 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C23×C4×C109 ◁ G | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,6,10,7,11,5,19)(3,12,16,22,15,28,14,8)(4,9,17,26,18,25,20,24)(13,29,30,23,31,27,32,21)(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);(1,3,5,14,7,15,6,16)(2,8,10,12,11,22,19,28)(4,13,17,30,18,31,20,32)(9,23,24,29,25,21,26,27);(1,4,5,17,7,18,6,20)(2,9,10,24,11,25,19,26)(3,13,14,30,15,31,16,32)(8,21,12,27,22,23,28,29)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4234);