Finite Group DatabasebetaFinite Group Databasebeta| order | 13696 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C107 · Schur multiplier (H2) = C22 |
| direct product | C8×Q16×C107 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C107 | D4 |
| commutator | C4 | C22×C8×C107 |
| Frattini, cosocle | C42 | C23×C107 |
| socle, prime radical | C22×C107 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C107 ◁ C4×C8×C107 ◁ G | |
164 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 13696 |
A minimal faithful permutation representation — 3 generators on 131 points:
(1,2,3,5,4,6,7,8)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)5992 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C107×#7069, C107×#7071, C8×SD32×C107, … |
| maximal normal subgroup of | C107×#7069, C107×#7071, C8×SD32×C107, … |
| maximal quotient of | C8×#103×C107, C8×#107×C107, C107×#8844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13696, 309);