Finite Group DatabasebetaFinite Group Databasebeta| order | 13184 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C23×C4 |
| direct product | C103×#3135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C103 |
| socle | C22×C103 | C22×D4 |
| prime radical | #532×C103 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C2×C42×C103 ◁ G | |
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,22)(13,18)(14,23)(15,19)(16,20)(21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,4,7)(2,5,6,8)(9,11,14,20,15,21,13,12)(10,16,18,17,19,22,23,24);(2,6)(5,8)(9,12,14,11,15,20,13,21)(10,17,18,22,19,24,23,16);(9,13,15,14)(10,18,19,23)(11,12,21,20)(16,17,22,24)3296 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C103×#9005, C103×#9008, C103×#10033, … |
| maximal normal subgroup of | C103×#9005, C103×#9008, C103×#10033, … |
| maximal quotient of | C103×#20762, C103×#21086, C103×#21199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13184, 1867);