Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C22 |
| direct product | #2155×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | #106 |
| commutator | C8 | C22×C4×C137 |
| Frattini, cosocle | C2×C8 | C23×C137 |
| socle | C2×C137 | C2×#106 |
| prime radical | C23×C137 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C22×C4×C137 ◁ C22×C8×C137 ◁ G | |
144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 169 points:
(1,2)(3,12,15,26)(4,9)(5,18)(6,21)(7,11)(8,14,23,28)(10,19)(13,25,16,30)(17,24)(20,31)(22,27,32,29)(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);(1,3,5,14,6,16,19,29,7,15,20,28,4,13,17,27)(2,8,10,12,9,22,18,25,11,23,24,26,21,32,31,30);(1,4,7,6)(2,9,11,21)(3,13,15,16)(5,17,20,19)(8,22,23,32)(10,18,24,31)(12,25,26,30)(14,27,28,29)4384 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#12668, C137×#12678, C137×#12832, … |
| maximal normal subgroup of | C137×#12668, C137×#12678, C137×#12832, … |
| maximal quotient of | C137×#11999, C137×#12000, C137×#12019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 887);