Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#12957 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | #442 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| prime radical | C23×C109 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×#97×C109 ◁ G | |
396 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)(5,6,10,17,12,18,11,29)(7,19,22,35,24,33,23,30)(8,16,26,34,27,13,9,15)(14,28,31,21,32,20,36,25)(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);(2,4)(6,13,18,16)(8,9)(10,11)(14,30,32,35)(15,17,34,29)(19,31,33,36)(20,21)(22,23)(25,28)(26,27);(5,7,12,24)(6,14,18,32)(8,25,27,21)(9,28,26,20)(10,23,11,22)(13,30,16,35)(15,33,34,19)(17,36,29,31)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5945);