Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C23 |
| direct product | #1985×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C137 | C23 |
| commutator | C23 | C22×C4×C137 |
| Frattini, cosocle | C22×C4 | C23×C137 |
| socle, prime radical | C23×C137 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C137 ◁ G | |
280 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 3 generators on 157 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19,17,20,16,15)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157);(1,2,3,4)(6,9)(10,11)(13,15,16,20,17,19,18,14);(5,7,8,12)(6,10,9,11)(13,16,17,18)(14,19,20,15)6028 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#7061, C137×#7133, C137×#7284, … |
| maximal normal subgroup of | C137×#7061, C137×#7133, C137×#7284, … |
| maximal quotient of | C137×#9867, C137×#9868, C137×#9869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 717);