Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C137 · Schur multiplier (H2) = C2 |
| direct product | C137×#7368 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | #354 |
| commutator | C42 | C2×C8×C137 |
| Frattini, cosocle | #423 | C22×C137 |
| socle | C2×C137 | #1324 |
| prime radical | C23×C137 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C2×C4×C137 ◁ C22×C4×C137 ◁ #423×C137 ◁ G | |
136 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 169 points:
(1,2,4,10,6,11,20,28,7,12,19,29,5,9,21,26)(3,13,14,27,16,8,22,23,17,24,30,32,15,25,18,31)(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);(1,3,5,15,7,17,6,16)(2,8,11,24,12,25,9,13)(4,18,20,14,19,22,21,30)(10,27,26,31,29,32,28,23)4658 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 356);