Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C24 |
| direct product | #1891×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | #120 |
| commutator | C23 | C22×C4×C137 |
| Frattini, cosocle | C22×C4 | C23×C137 |
| socle | C2×C137 | #386 |
| prime radical | #525×C137 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C22×C4×C137 ◁ G | |
342 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 153 |
| transitive degree | μt(G) = 2192 |
A minimal faithful permutation representation — 3 generators on 153 points:
(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)4384 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#12755, C137×#12756, C137×#12757, … |
| maximal normal subgroup of | C137×#12755, C137×#12756, C137×#12757, … |
| maximal quotient of | C137×#8965, C137×#8966, C137×#8967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 623);