Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C97 · Schur multiplier (H2) = C22 |
| direct product | C2×#368×C97 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C97 | D8 |
| commutator | C8 | C22×C4×C97 |
| Frattini, cosocle | C2×C8 | C23×C97 |
| socle | C22×C97 | #107 |
| prime radical | C23×C97 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C97 ◁ C22×C4×C97 ◁ C22×C8×C97 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 131 points:
(1,2)(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);(3,4,7,12,9,13,8,23)(5,14,17,29,19,30,18,33)(6,20,21,27,22,34,15,11)(10,24,25,32,26,31,28,16);(3,5,15,31,8,18,22,32,9,19,21,24,7,17,6,16)(4,10,20,29,12,25,27,30,13,26,34,33,23,28,11,14)4268 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 892);