Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C97 · Schur multiplier (H2) = C22 |
| direct product | C97×C22⋊C32 |
| unambiguous semidirect product | C22⋊(C32×C97) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C97 | C22 |
| commutator | C2 | C2×C32×C97 |
| Frattini, cosocle | C2×C16 | C22×C97 |
| socle | C22×C97 | C2×C16 |
| prime radical | C23×C97 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C97 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2)(3,4)(5,6,7,11,8,12,15,21,9,13,16,22,18,24,27,31,10,14,17,23,19,25,28,32,20,26,29,33,30,34,35,36)(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);(2,4)7760 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 131);