Finite Group DatabasebetaFinite Group Databasebeta| order | 13696 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C107 · Schur multiplier (H2) = C24 |
| direct product | C107×#1595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C107 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ G | |
368 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(2,6)(3,9)(7,13)(8,10)(11,15)(14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,2)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15);(3,11)(6,13)(9,15)(10,16)2461 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C107×#12090, C107×#12092, C107×#12093, … |
| maximal normal subgroup of | C107×#12090, C107×#12092, C107×#12093, … |
| maximal quotient of | C107×#7734, C107×#7735, C107×#7736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13696, 327);