Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C137 · Schur multiplier (H2) = C22 |
| direct product | #1392×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | #99 |
| commutator | C23 | C42×C137 |
| Frattini, cosocle | C2×#41 | C22×C137 |
| socle | C2×C137 | #342 |
| prime radical | C2×#41×C137 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C2×C4×C137 ◁ C2×#41×C137 ◁ G | |
158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 169 points:
(1,2,5,10,7,12,6,11)(3,13,15,28,17,30,16,29)(4,18,19,27,20,9,21,25)(8,22,23,14,24,31,26,32)(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);(1,3,6,16,7,17,5,15)(2,8,11,23,12,24,10,26)(4,14,19,31,20,32,21,22)(9,13,25,29,18,30,27,28)3562 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#7400, C137×#7401, C137×#7407, … |
| maximal normal subgroup of | C137×#7400, C137×#7401, C137×#7407, … |
| maximal quotient of | C137×#7188, C137×#7203, C137×#7216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 124);