Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C109 · Schur multiplier (H2) = C24 |
| direct product | C2×C109×#1385 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | #106 |
| commutator | C8 | C2×C42×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| socle | C22×C109 | #340 |
| prime radical | C24×C109 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C23×C4×C109 ◁ C23×C8×C109 ◁ G | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(3,4,7,13,10,15,9,25)(5,16,18,31,20,32,23,27)(6,14,8,24,22,30,21,12)(11,19,26,17,29,33,28,34);(3,5,8,19,7,23,22,34,10,20,21,33,9,18,6,17)(4,11,14,27,25,26,12,16,15,29,30,31,13,28,24,32)6976 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5553);