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 | #2214×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C137 |
| prime radical | #135×C137 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C2×C4×C137 ◁ C4×C8×C137 ◁ G | |
150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 150 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:
(1,2,8,14)(3,12,17,23)(4,18,20,31)(5,13,7,11)(6,9,16,24)(10,25,27,15)(19,32,22,28)(21,29,30,26)(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);(1,3,7,16,8,17,5,6)(2,9,13,23,14,24,11,12)(4,15,22,30,20,25,19,21)(10,18,29,32,27,31,26,28);(1,4)(2,10)(3,15)(5,19)(6,21)(7,22)(8,20)(9,18)(11,26)(12,28)(13,29)(14,27)(16,30)(17,25)(23,32)(24,31)3151 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#12312, C137×#12313, C137×#12315, … |
| maximal normal subgroup of | C137×#12312, C137×#12313, C137×#12315, … |
| maximal quotient of | C137×#12422, C137×#12423, C137×#12426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 946);