Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#12997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | #447 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| prime radical | C2×#41×C109 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×#131×C109 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 3 generators on 145 points:
(2,4)(5,6,10,17,9,16,25,13,12,19,8,15,11,18,26,29)(7,20,23,28,27,14,30,32,24,34,21,35,22,33,36,31)(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);(1,2,3,4)(6,13)(8,25)(9,11)(10,26)(14,28)(15,18)(16,17)(19,29)(20,32)(21,30)(22,27)(23,36)(31,34)(33,35);(5,7)(6,14)(8,23)(9,27)(10,21)(11,22)(12,24)(13,28)(15,32)(16,34)(17,31)(18,20)(19,33)(25,36)(26,30)(29,35)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5985);