Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C22 |
| direct product | #2180×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #442 |
| commutator | C8 | C22×C4×C137 |
| Frattini, cosocle | C2×C8 | C23×C137 |
| prime radical | #135×C137 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C2×C4×C137 ◁ #97×C137 ◁ G | |
142 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,13,15,24)(4,17)(5,9)(6,21)(7,12)(8,14,25,31)(10,18)(11,23)(16,28,22,30)(19,29,32,26)(20,27)(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)(2,8)(4,14)(5,19)(6,22)(7,15)(9,24)(10,28)(11,29)(12,25)(13,27)(16,23)(17,30)(18,31)(20,32)(21,26);(1,4,6,5,7,18,23,20)(2,9,11,17,12,27,21,10)(3,14,22,19,15,31,16,32)(8,24,29,30,25,13,26,28)3151 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#2180×C137, C137×#33468, C137×#33469, … |
| maximal normal subgroup of | C2×#2180×C137, C137×#33468, C137×#33469, … |
| maximal quotient of | C137×#12065, C137×#12066, C137×#12077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 912);