Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C97 · Schur multiplier (H2) = C2 |
| direct product | C97×#1397 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C97 | C22 |
| commutator | C2 | C4×C16×C97 |
| Frattini, cosocle | C2×C16 | C22×C97 |
| socle, prime radical | C22×C97 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C97 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 2 generators on 133 points:
(5,6,7,12,8,13,15,24,9,14,16,25,18,26,27,34,10,11,17,21,19,22,28,31,20,23,29,32,30,33,35,36)(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);(1,2,3,4)(6,11)(12,21)(13,22)(14,23)(24,31)(25,32)(26,33)(34,36)7760 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 129);