Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C137 · Schur multiplier (H2) = C23 |
| direct product | C137×#12168 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C137 |
| prime radical | C23×C137 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C22×C4×C137 ◁ #424×C137 ◁ G | |
330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 209 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 3 generators on 209 points:
(5,8)(9,10,16,22)(11,23,27,51)(12,28,31,48)(13,32,36,46)(14,21,15,20)(17,39,42,25)(18,37,44,30)(19,34,47,35)(24,52,56,69)(26,57,38,45)(29,60,63,43)(33,49,64,41)(40,61,68,72)(50,55,71,54)(53,58,66,67)(59,65,70,62)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209);(1,2)(3,5)(4,6)(7,8)(9,11,16,27)(10,17,22,42)(12,24,31,56)(13,33,36,64)(14,26,15,38)(18,40,44,68)(19,45,47,57)(20,41,21,49)(23,46,51,32)(25,34,39,35)(28,58,48,67)(29,61,63,72)(30,54,37,55)(43,69,60,52)(50,70,71,59)(53,65,66,62);(1,3,4,7)(2,5,6,8)(9,12)(10,18)(11,24)(13,29)(14,37)(15,30)(16,31)(17,40)(19,43)(20,48)(21,28)(22,44)(23,50)(25,53)(26,55)(27,56)(32,59)(33,61)(34,65)(35,62)(36,63)(38,54)(39,66)(41,67)(42,68)(45,69)(46,70)(47,60)(49,58)(51,71)(52,57)(64,72)4247 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 5156);