Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C97 · Schur multiplier (H2) = C27 |
| direct product | C22×#99×C97 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C97 | C22⋊C4 |
| commutator | C22 | C23×C4×C97 |
| Frattini, cosocle | C23 | C24×C97 |
| prime radical | C23×D4×C97 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C97 ◁ C25×C97 ◁ G | |
1112 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 4 generators on 109 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)(104,108,109,106)(105,107);(102,104)(103,106)(105,109)(107,108)4268 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 1613);