Finite Group DatabasebetaFinite Group Databasebeta| order | 13696 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C107 · Schur multiplier (H2) = C2 |
| direct product | C107×#1299 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C107 | C22⋊C4 |
| commutator | C2×C4 | C42×C107 |
| Frattini, cosocle | C2×C42 | C22×C107 |
| socle | C22×C107 | #95 |
| prime radical | C23×C107 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C107 ◁ C2×C42×C107 ◁ G | |
150 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 6848 |
A minimal faithful permutation representation — 2 generators on 123 points:
(1,2,4,7,5,8,3,6)(10,13)(11,15)(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);(2,6,8,7)(9,10,11,14,12,13,15,16)4066 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C107×#1299, C107×#8060, C107×#8061, … |
| maximal normal subgroup of | C2×C107×#1299, C107×#8060, C107×#8061, … |
| maximal quotient of | C107×#7026, C107×#7030, C107×#7031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13696, 31);