Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C137 · Schur multiplier (H2) = C25 |
| direct product | #3293×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C137 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C2×C4×C137 ◁ G | |
444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 169 points:
(1,2,8,13)(3,14,17,19)(4,16,9,23)(5,10,22,28)(6,12,7,11)(15,30,32,26)(18,31,25,20)(21,29,24,27)(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);(1,3,8,17)(2,9,13,4)(5,20,22,31)(6,23,7,16)(10,26,28,30)(11,19,12,14)(15,24,32,21)(18,27,25,29);(1,4,8,9)(2,3,13,17)(5,18,22,25)(6,19,7,14)(10,15,28,32)(11,16,12,23)(20,29,31,27)(21,30,24,26);(1,5)(2,10)(3,15)(4,18)(6,24)(7,21)(8,22)(9,25)(11,29)(12,27)(13,28)(14,30)(16,31)(17,32)(19,26)(20,23)3973 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#12311, C137×#12313, C137×#12367, … |
| maximal normal subgroup of | C137×#12311, C137×#12313, C137×#12367, … |
| maximal quotient of | C137×#19887, C137×#19894, C137×#19902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 2025);