Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C137 · Schur multiplier (H2) = C22 |
| direct product | #2212×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C137 |
| prime radical | #509×C137 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C2×C4×C137 ◁ C4×C8×C137 ◁ G | |
154 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,11)(4,16)(5,20)(6,22)(7,12)(8,14)(9,13)(10,21)(15,29)(17,28)(18,32)(19,24)(23,25)(26,30)(27,31)(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);(1,3,5,14,7,15,21,6)(2,8,10,11,12,22,20,29)(4,13,17,30,19,31,18,23)(9,24,25,28,27,16,26,32);(1,4)(2,9)(3,13)(5,17)(6,23)(7,19)(8,24)(10,25)(11,28)(12,27)(14,30)(15,31)(16,22)(18,21)(20,26)(29,32)3151 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#2212×C137, C137×#33594, C137×#33595, … |
| maximal normal subgroup of | C2×#2212×C137, C137×#33594, C137×#33595, … |
| maximal quotient of | C137×#12416, C137×#12417, C137×#12420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 944);