Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C109 · Schur multiplier (H2) = C24×C4 |
| direct product | C109×#20487 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C109 | C24 |
| commutator | C22 | C22×C42×C109 |
| Frattini, cosocle | C22×C4 | C24×C109 |
| socle | C23×C109 | C23×C4 |
| prime radical | C22×D4×C109 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C109 ◁ G | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 4 generators on 133 points:
(1,2)(3,4)(9,10,13,12,14,16,21,20)(11,15,19,18,17,22,24,23)(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);(2,4)(5,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24);(5,7,8,6)(11,17)(15,22)(18,23)(19,24);(9,12,13,16,14,20,21,10)(11,18,19,22,17,23,24,15)9592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 13475);