Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C137 · Schur multiplier (H2) = C25 |
| direct product | #3415×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ #41×C137 ◁ #131×C137 ◁ G | |
370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 370 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) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2)(3,8)(4,7)(5,18)(6,11)(9,15)(10,16)(12,22)(13,19)(14,28)(17,21)(20,29)(23,27)(24,26)(25,30)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169);(2,7)(4,15)(5,16)(8,19)(9,10)(11,21)(13,27)(14,26)(17,29)(18,20)(22,30)(23,24)(25,32)(28,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,23)(10,24)(11,22)(15,27)(16,26)(17,25)(18,28)(20,31)(21,30)(29,32);(3,12)(8,22)(13,25)(14,26)(19,30)(23,31)(24,28)(27,32)4384 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#13669, C137×#13671, C137×#33854, … |
| maximal normal subgroup of | C137×#13669, C137×#13671, C137×#33854, … |
| maximal quotient of | C137×#33381, C137×#33410, C137×#33454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 2147);