Finite Group DatabasebetaFinite Group Databasebeta| order | 12608 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 2023096320 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C210 |
| abelian | C24×C4×C197 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C25×C197 |
| socle, prime radical | C25×C197 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 209 |
| transitive degree | μt(G) = 12608 |
A minimal faithful permutation representation — 5 generators on 209 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209);(5,6);(7,8);(9,10);(11,12)12608 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12608, 260);