Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#12984 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C109 | C2×D4 |
| commutator | C2×C4 | C22×C8×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| socle | C22×C109 | C2×C22⋊C8 |
| prime radical | #525×C109 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C109 ◁ C23×C8×C109 ◁ G | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 145 points:
(1,2)(3,4)(5,6,9,14,10,13,23,26,11,15,24,29,8,22,17,32)(7,16,20,31,21,27,34,35,18,30,33,36,19,12,25,28)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(5,7,17,25,8,19,24,33,11,18,23,34,10,21,9,20)(6,12,26,31,22,30,14,28,15,27,32,36,13,16,29,35);(2,4)(7,18)(12,27)(16,30)(19,21)(20,33)(25,34)(28,35)(31,36)6976 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5972);