Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#12963 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C109 |
| Frattini, cosocle | C2×#41 | C23×C109 |
| socle | C2×C109 | #1867 |
| prime radical | #525×C109 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C2×C4×C109 ◁ C2×#41×C109 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 3488 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,5,9,6,10,17,23)(3,11,14,25,12,24,27,32)(4,8,16,22,15,21,28,31)(7,18,20,26,19,29,30,13)(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);(1,3,5,14,6,12,17,27)(2,7,9,20,10,19,23,30)(4,13,16,18,15,26,28,29)(8,11,22,25,21,24,31,32);(3,12)(4,15)(7,19)(11,24)(14,27)(16,28)(20,30)(25,32)3706 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5951);