Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C137 · Schur multiplier (H2) = C24 |
| direct product | #3202×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C137 |
| prime radical | C2×#41×C137 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C2×#41×C137 ◁ G | |
378 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 177 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 4 generators on 177 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,20,29)(12,22,24,32)(13,25,27,17)(16,26,31,23)(19,33,34,40)(21,37,36,30)(28,38,39,35)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177);(1,3)(2,5)(4,8)(6,7)(9,11,14,20)(10,15,18,29)(12,19,24,34)(13,21,27,36)(16,28,31,39)(17,30,25,37)(22,33,32,40)(23,35,26,38);(3,8)(5,7)(9,12,14,24)(10,16,18,31)(11,19,20,34)(13,26,27,23)(15,28,29,39)(17,32,25,22)(21,38,36,35)(30,40,37,33);(2,6)(5,7)(11,20)(15,29)(19,34)(21,36)(28,39)(30,37)(33,40)(35,38)3562 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#12109, C137×#12112, C137×#12118, … |
| maximal normal subgroup of | C137×#12109, C137×#12112, C137×#12118, … |
| maximal quotient of | C137×#21771, C137×#21947, C137×#22054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 1934);