Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#12959 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C109 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C109 |
| Frattini, cosocle | C22×D4 | C23×C109 |
| prime radical | C22×D4×C109 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C23×C109 ◁ C22×D4×C109 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,17)(14,19)(15,20)(16,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,3,4,2)(6,10)(11,17)(12,18)(19,20);(5,7,9,14)(6,11,12,19)(8,15,16,13)(10,17,18,20)4360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5947);