Finite Group DatabasebetaFinite Group Databasebeta| order | 13696 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C107 · Schur multiplier (H2) = C23 |
| direct product | C107×#1282 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | C22⋊C4 |
| commutator | C22 | C4×C8×C107 |
| Frattini, cosocle | C23×C4 | C22×C107 |
| socle | C22×C107 | #95 |
| prime radical | C24×C107 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C23×C4×C107 ◁ G | |
228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 3424 |
A minimal faithful permutation representation — 2 generators on 123 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,15,14,16,13,11)(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);(1,3)(2,5)(4,8)(6,7)(9,11,13,16,14,15,12,10)4708 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C107×#1282, C107×#7924, C107×#7925, … |
| maximal normal subgroup of | C2×C107×#1282, C107×#7924, C107×#7925, … |
| maximal quotient of | C107×#7016, C107×#7021, C107×#7024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13696, 14);