Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#11254 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #394 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C23×C4 | C23×C109 |
| socle | C2×C109 | #1441 |
| prime radical | C24×C109 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C23×C4×C109 ◁ G | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,6,10,7,11,5,22)(3,12,15,25,16,24,14,8)(4,17,20,28,21,27,19,9)(13,23,29,26,30,31,18,32)(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);(1,3,5,14,7,16,6,15)(2,8,10,12,11,25,22,24)(4,18,19,30,21,29,20,13)(9,26,17,31,28,32,27,23);(1,4,5,19,7,21,6,20)(2,9,10,17,11,28,22,27)(3,13,14,18,16,30,15,29)(8,23,12,26,25,31,24,32)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4242);