metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D40⋊7C2, C4.20D20, C20.35D4, C8.17D10, Dic20⋊7C2, C22.1D20, C40.17C22, C20.30C23, D20.7C22, Dic10.6C22, (C2×C8)⋊4D5, (C2×C40)⋊6C2, C5⋊1(C4○D8), C4○D20⋊1C2, C40⋊C2⋊7C2, C2.13(C2×D20), C10.11(C2×D4), (C2×C10).18D4, (C2×C4).81D10, C4.28(C22×D5), (C2×C20).99C22, SmallGroup(160,125)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40⋊7C2
G = < a,b,c | a40=b2=c2=1, bab=a-1, ac=ca, cbc=a20b >
Subgroups: 232 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, C2×C10, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C40⋊C2, D40, Dic20, C2×C40, C4○D20, D40⋊7C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4○D8, D20, C22×D5, C2×D20, D40⋊7C2
►On 80 points
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 51)(42 50)(43 49)(44 48)(45 47)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,51)(42,50)(43,49)(44,48)(45,47)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,51),(42,50),(43,49),(44,48),(45,47),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)]])
D40⋊7C2 is a maximal subgroup of
D40⋊14C4 D40.5C4 D40.3C4 D40⋊8C4 D40⋊17C4 D40⋊10C4 D40⋊16C4 D40⋊13C4 D80⋊7C2 D80⋊C2 C16.D10 D8.D10 Q16.D10 C40.30C23 C40.9C23 D4.11D20 D4.12D20 D4.13D20 D8⋊13D10 D20.29D4 D20.30D4 D5×C4○D8 Q16⋊D10 D6.1D20 D40⋊7S3 D120⋊5C2 D20.31D6 C40.69D6
D40⋊7C2 is a maximal quotient of
C40.13Q8 C4×C40⋊C2 C4×D40 C8.8D20 C42.264D10 C4×Dic20 C23.10D20 D20.32D4 D20⋊14D4 C23.13D20 Dic10.3Q8 D20.19D4 C42.36D10 D20.3Q8 C23.22D20 C23.23D20 C40⋊30D4 C40⋊29D4 C40.82D4 D6.1D20 D40⋊7S3 D120⋊5C2 D20.31D6 C40.69D6
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 1 | 1 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 |
| kernel | D40⋊7C2 | C40⋊C2 | D40 | Dic20 | C2×C40 | C4○D20 | C20 | C2×C10 | C2×C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 16 |
Matrix representation of D40⋊7C2 ►in GL2(𝔽41) generated by
| 26 | 21 |
| 23 | 6 |
| 27 | 5 |
| 2 | 14 |
| 24 | 34 |
| 6 | 17 |
G:=sub<GL(2,GF(41))| [26,23,21,6],[27,2,5,14],[24,6,34,17] >;
D40⋊7C2 in GAP, Magma, Sage, TeX
D_{40}\rtimes_7C_2 % in TeX
G:=Group("D40:7C2"); // GroupNames label
G:=SmallGroup(160,125);
// by ID
G=gap.SmallGroup(160,125);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,50,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^20*b>;
// generators/relations