Finite Group DatabasebetaFinite Group Databasebeta| order | 25216 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C197 · Schur multiplier (H2) = C2 |
| direct product | #2253×C197 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C197 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C197 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C197 ◁ C42×C197 ◁ C4×C8×C197 ◁ G | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 237 |
| transitive degree | μt(G) = 25216 |
A minimal faithful permutation representation — 3 generators on 237 points:
(1,2,4,6)(3,7,8,5)(9,10,12,15)(11,19,21,33)(13,26,24,39)(14,27,25,18)(16,30,28,22)(17,32,29,20)(23,38,37,35)(31,36,34,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237);(1,3,4,8)(2,5,6,7)(10,15)(16,28)(17,29)(18,27)(19,33)(26,39)(31,34)(35,38);(9,11,20,36,14,23,24,30,12,21,32,40,25,37,13,22)(10,16,26,35,18,31,29,33,15,28,39,38,27,34,17,19)7486 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25216, 985);