Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C137 · Schur multiplier (H2) = C23 |
| direct product | #2194×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C137 |
| prime radical | #576×C137 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C2×C4×C137 ◁ #131×C137 ◁ G | |
216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,7,13)(3,14,17,30)(4,18,22,29)(5,12,6,11)(8,24,27,15)(9,23,28,16)(10,20,25,21)(19,31,32,26)(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)(2,8)(4,19)(5,16)(6,23)(7,17)(9,25)(10,28)(11,26)(12,31)(13,27)(14,18)(15,21)(20,24)(22,32)(29,30);(1,3)(2,9)(4,15)(5,23)(6,16)(7,17)(8,25)(10,27)(11,30)(12,14)(13,28)(18,31)(19,21)(20,32)(22,24)(26,29)3151 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#2194×C137, C137×#33538, C137×#33539, … |
| maximal normal subgroup of | C2×#2194×C137, C137×#33538, C137×#33539, … |
| maximal quotient of | C137×#12111, C137×#12113, C137×#12167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 926);