Finite Group DatabasebetaFinite Group Databasebeta| order | 13184 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#3414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C103 |
| socle | C2×C103 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ C22×C8×C103 ◁ G | |
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,12)(4,9)(5,19)(6,8)(7,14)(10,17)(11,18)(13,26)(15,31)(16,28)(20,27)(21,25)(22,30)(23,29)(24,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,3)(2,7)(4,13)(5,20)(6,16)(8,22)(9,23)(10,26)(11,27)(12,19)(14,17)(15,18)(21,29)(24,31)(25,28)(30,32);(2,8)(7,22)(9,11)(10,24)(12,28)(19,25)(23,27)(26,31);(1,4,6,18)(2,9,8,11)(3,13,16,15)(5,17,21,32)(7,23,22,27)(10,25,24,19)(12,26,28,31)(14,29,30,20)3296 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C103×#13021, C103×#13024, C103×#13027, … |
| maximal normal subgroup of | C103×#13021, C103×#13024, C103×#13027, … |
| maximal quotient of | C103×#33380, C103×#33409, C103×#33451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13184, 2146);