metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5C4, C2.2D40, C10.5D8, C20.45D4, C10.3SD16, C22.10D20, (C2×C8)⋊2D5, (C2×C40)⋊2C2, C4.8(C4×D5), C4⋊Dic5⋊1C2, C5⋊3(D4⋊C4), C20.39(C2×C4), (C2×D20).1C2, (C2×C4).71D10, (C2×C10).15D4, C2.3(C40⋊C2), C4.20(C5⋊D4), (C2×C20).83C22, C2.8(D10⋊C4), C10.18(C22⋊C4), SmallGroup(160,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊5C4
G = < a,b,c | a20=b2=c4=1, bab=cac-1=a-1, cbc-1=a3b >
Smallest permutation representation of D20⋊5C4
►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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 41)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)
(1 58 71 30)(2 57 72 29)(3 56 73 28)(4 55 74 27)(5 54 75 26)(6 53 76 25)(7 52 77 24)(8 51 78 23)(9 50 79 22)(10 49 80 21)(11 48 61 40)(12 47 62 39)(13 46 63 38)(14 45 64 37)(15 44 65 36)(16 43 66 35)(17 42 67 34)(18 41 68 33)(19 60 69 32)(20 59 70 31)
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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42), (1,58,71,30)(2,57,72,29)(3,56,73,28)(4,55,74,27)(5,54,75,26)(6,53,76,25)(7,52,77,24)(8,51,78,23)(9,50,79,22)(10,49,80,21)(11,48,61,40)(12,47,62,39)(13,46,63,38)(14,45,64,37)(15,44,65,36)(16,43,66,35)(17,42,67,34)(18,41,68,33)(19,60,69,32)(20,59,70,31)>;
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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42), (1,58,71,30)(2,57,72,29)(3,56,73,28)(4,55,74,27)(5,54,75,26)(6,53,76,25)(7,52,77,24)(8,51,78,23)(9,50,79,22)(10,49,80,21)(11,48,61,40)(12,47,62,39)(13,46,63,38)(14,45,64,37)(15,44,65,36)(16,43,66,35)(17,42,67,34)(18,41,68,33)(19,60,69,32)(20,59,70,31) );
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)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,41),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42)], [(1,58,71,30),(2,57,72,29),(3,56,73,28),(4,55,74,27),(5,54,75,26),(6,53,76,25),(7,52,77,24),(8,51,78,23),(9,50,79,22),(10,49,80,21),(11,48,61,40),(12,47,62,39),(13,46,63,38),(14,45,64,37),(15,44,65,36),(16,43,66,35),(17,42,67,34),(18,41,68,33),(19,60,69,32),(20,59,70,31)]])
D20⋊5C4 is a maximal subgroup of
C4×C40⋊C2 C4×D40 C4.5D40 C42.264D10 C42.16D10 D40⋊9C4 C42.19D10 C42.20D10 D20.31D4 D20⋊13D4 D20.32D4 D20⋊14D4 C23.38D20 C22.D40 C23.13D20 Dic5⋊4D8 Dic5.5D8 C4⋊C4.D10 D5×D4⋊C4 (D4×D5)⋊C4 D10⋊D8 C5⋊(C8⋊2D4) D4⋊D5⋊6C4 Dic5⋊7SD16 Q8⋊C4⋊D5 Q8⋊Dic5⋊C2 Q8⋊(C4×D5) Q8⋊2D5⋊C4 D10⋊2SD16 C5⋊(C8⋊D4) Q8⋊D5⋊6C4 D20⋊3Q8 C4⋊D40 D20.19D4 C42.36D10 D20⋊4Q8 D20.3Q8 Dic10⋊8D4 D10.17SD16 C4.Q8⋊D5 D20⋊Q8 D20.Q8 D10.13D8 C2.D8⋊D5 D20⋊2Q8 D20.2Q8 C23.23D20 C40⋊30D4 C40⋊29D4 C23.48D20 C23.49D20 C40⋊2D4 C40⋊3D4 Dic5⋊D8 D20⋊D4 Dic5⋊5SD16 (C5×D4).D4 D10⋊6SD16 D20⋊7D4 (C2×Q16)⋊D5 D20.17D4 C6.D40 D60⋊12C4 D60⋊8C4
D20⋊5C4 is a maximal quotient of
D20⋊3C8 C22.2D40 C4.D40 C20.2D8 C40.78D4 D40⋊7C4 D40.3C4 D40.4C4 C20.4D8 D40⋊8C4 C20.39C42 C6.D40 D60⋊12C4 D60⋊8C4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4×D5 | C5⋊D4 | D20 | C40⋊C2 | D40 |
| kernel | D20⋊5C4 | C4⋊Dic5 | C2×C40 | C2×D20 | D20 | C20 | C2×C10 | C2×C8 | C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of D20⋊5C4 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 25 | 11 |
| 0 | 14 | 39 |
| 1 | 0 | 0 |
| 0 | 2 | 11 |
| 0 | 37 | 39 |
| 9 | 0 | 0 |
| 0 | 14 | 28 |
| 0 | 12 | 27 |
G:=sub<GL(3,GF(41))| [1,0,0,0,25,14,0,11,39],[1,0,0,0,2,37,0,11,39],[9,0,0,0,14,12,0,28,27] >;
D20⋊5C4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_5C_4 % in TeX
G:=Group("D20:5C4"); // GroupNames label
G:=SmallGroup(160,28);
// by ID
G=gap.SmallGroup(160,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,73,79,362,86,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^3*b>;
// generators/relations
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