Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C137 · Schur multiplier (H2) = C2×C4 |
| direct product | #1564×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C137 | D4 |
| commutator | C4 | C22×C8×C137 |
| Frattini, cosocle | C42 | C23×C137 |
| socle | C22×C137 | C2×C4⋊C4 |
| prime radical | C23×C137 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C137 ◁ C2×C42×C137 ◁ G | |
180 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 161 points:
(1,2,3,5,4,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161);(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)7672 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#7155, C137×#7157, C137×#7158, … |
| maximal normal subgroup of | C137×#7155, C137×#7157, C137×#7158, … |
| maximal quotient of | C137×#7872, C4×#334×C137, C4×#335×C137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 296);