Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C109 · Schur multiplier (H2) = C22×C4 |
| direct product | C109×#8239 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C109 |
| Frattini, cosocle | C23×C4 | C23×C109 |
| socle | C22×C109 | C4×C22⋊C4 |
| prime radical | #525×C109 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C23×C8×C109 ◁ G | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 133 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,18,12,15)(11,16,21,23,19,24,20,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,3)(2,5,6,7)(4,8)(10,15,18,17)(11,19)(16,23,24,22)(20,21);(9,11,13,21,14,19,12,20)(10,16,17,23,18,24,15,22)5668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 1227);