Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C22×C8 |
| direct product | C2×C8×#11715906 |
| Fitting | C2×C8×C97 ⋊ C8 |
| derived, Sylow | C97 ⋊ C2×C82 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | #11715906 |
| commutator, hypocenter | C97 | C2×C82 |
| Frattini | C4 | C22×#11715906 |
| Fitting | C2×C8×C97 | C8 |
| socle | C22×C97 | C4×C8 |
| prime radical | C22×D97 | C42 |
| cosocle | C4×#87604 | C23 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C2×C8 | |
280 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 1552 |
A minimal faithful permutation representation — 3 generators on 107 points:
(2,3,4,7,5,8,10,15)(6,12,13,27,14,28,30,36)(9,18,19,39,20,40,42,32)(11,23,24,49,25,50,22,47)(16,33,34,60,35,17,37,38)(21,44,45,67,46,70,26,52)(29,53,54,77,55,78,80,59)(31,57,58,82,43,69,48,73)(41,64,65,88,66,56,68,72)(51,75,76,87,71,63,89,62)(61,85,81,74,86,92,84,83)(79,94,95,96,93,97,90,91);(1,2,6,12,29,56,53,79,89,54,81,38,62,88,49,74,69,23,51,70,45,25,10,22,48,34,61,75,18,41,67,27,35,15,32,59,73,91,97,85,95,65,19,43,30,46,20,8,17,33,3,9,21,13,31,42,68,90,92,94,96,82,77,39,7,16,36,52,66,40,63,86,37,58,24,4,11,26,44,71,50,57,83,47,72,87,60,84,80,76,93,78,64,55,28,14,5)(100,101,102,104,103,105,106,107);(98,99)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21616);