Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C97 · Schur multiplier (H2) = C26 |
| direct product | C2×#408×C97 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C97 | C23 |
| commutator | C22 | C23×C4×C97 |
| Frattini, cosocle | C2×C4 | C24×C97 |
| socle | C23×C97 | C22×C4 |
| prime radical | C22×#41×C97 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C97 ◁ G | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97)(98,99);(100,101)(102,103)(104,105,107,110,108,111,114,117)(106,109,113,116,112,115,118,119);(101,103)(104,106)(105,109)(107,113)(108,112)(110,116)(111,115)(114,118)(117,119);(106,112)(109,115)(113,118)(116,119)5432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 1610);