Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C109 · Schur multiplier (H2) = C24 |
| direct product | C109×#12980 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C109 | C2×D4 |
| commutator | C2×C4 | C22×C8×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| socle | C22×C109 | C8×D4 |
| prime radical | C2×#135×C109 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C109 ◁ C22×C16×C109 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 145 points:
(2,4)(5,6,10,16,11,17,30,21,12,18,31,35,9,15,20,33)(7,19,14,34,22,8,24,26,23,25,28,36,27,13,32,29)(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)(5,7,20,24,9,27,31,14,12,23,30,32,11,22,10,28)(6,13,33,34,15,25,35,29,18,8,21,36,17,19,16,26);(5,8)(6,14)(7,21)(9,19)(10,29)(11,25)(12,13)(15,32)(16,27)(17,24)(18,28)(20,36)(22,35)(23,33)(26,31)(30,34)8284 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5968);