Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#12952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1864 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| prime radical | C2×#135×C109 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C22×C4×C109 ◁ #405×C109 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(2,7,10,9)(3,12,13,31)(4,15)(5,19)(8,23)(11,28,20,16)(14,17,21,32)(18,30)(22,24)(25,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,2,5,20,6,10,19,11)(3,8,21,26,13,23,14,29)(4,16,15,7,18,28,30,9)(12,27,17,25,31,22,32,24);(1,3)(2,8)(4,17)(5,21)(6,13)(7,22)(9,27)(10,23)(11,29)(12,30)(14,19)(15,31)(16,25)(18,32)(20,26)(24,28)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5940);