Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C137 · Schur multiplier (H2) = C2 |
| direct product | #2231×C137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #476 |
| 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 | |
120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 120 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:
(1,2,7,12)(3,13,16,11)(4,17)(5,21,14,10)(6,8,23,25)(9,26)(15,29)(18,31)(19,30)(20,24)(22,27)(28,32)(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);(1,3,14,23,7,16,5,6)(2,8,21,11,12,25,10,13)(4,15,19,32,20,26,18,22)(9,24,28,30,29,17,27,31);(1,4)(2,9)(3,15)(5,18)(6,22)(7,20)(8,24)(10,27)(11,30)(12,29)(13,31)(14,19)(16,26)(17,25)(21,28)(23,32)3151 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#12463, C137×#12464, C137×#12465, … |
| maximal normal subgroup of | C137×#12463, C137×#12464, C137×#12465, … |
| maximal quotient of | C137×#12824, C137×#12950, C137×#13116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 963);