Finite Group DatabasebetaFinite Group Databasebeta| order | 13696 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C107 · Schur multiplier (H2) = C23 |
| direct product | C107×#2048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C107 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C107 |
| prime radical | C2×AGL(1,ℤ/8)×C107 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C107 ◁ C22×C4×C107 ◁ G | |
274 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 1712 |
A minimal faithful permutation representation — 3 generators on 123 points:
(2,7)(3,10)(4,5)(8,15)(9,13)(11,12)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,2)(3,8)(4,13)(5,7)(6,9)(10,14)(11,16)(12,15);(1,3,4,11,6,12,5,10)(2,8,7,14,9,15,13,16)2461 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C107×#2048, C107×#23525, C107×#24295, … |
| maximal normal subgroup of | C2×C107×#2048, C107×#23525, C107×#24295, … |
| maximal quotient of | C107×#10440, C107×#10441, C107×#10442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13696, 780);