Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C137 · Schur multiplier (H2) = C2 |
| direct product | C137×#7446 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | #357 |
| commutator | C2×C16 | C2×C4×C137 |
| Frattini, cosocle | C4×C16 | C22×C137 |
| prime radical | C2×D16×C137 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C42×C137 ◁ C4×C8×C137 ◁ C4×C16×C137 ◁ G | |
136 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 177 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 2 generators on 177 points:
(1,2,3,6,4,5,7,8)(10,16,17,32)(11,21)(12,27)(13,23)(14,29)(18,35,19,36)(20,37,31,34)(22,28,33,30)(24,26)(25,40)(38,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177);(2,5)(3,7)(9,10)(11,22)(12,28)(13,30)(14,31)(15,17)(16,24)(18,27)(19,23)(20,29)(21,32)(25,35)(26,33)(34,39)(36,38)(37,40)6713 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 434);