metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊2D4, D10⋊3D4, C23.8D10, (C2×D4)⋊4D5, (D4×C10)⋊3C2, C4⋊2(C5⋊D4), C5⋊4(C4⋊D4), C2.26(D4×D5), C4⋊Dic5⋊14C2, (C2×C4).51D10, C10.50(C2×D4), C23.D5⋊11C2, C10.31(C4○D4), (C2×C10).53C23, (C2×C20).34C22, C2.17(D4⋊2D5), C22.60(C22×D5), (C22×C10).20C22, (C2×Dic5).19C22, (C22×D5).29C22, (C2×C4×D5)⋊2C2, (C2×C5⋊D4)⋊5C2, C2.14(C2×C5⋊D4), SmallGroup(160,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊2D4
G = < a,b,c | a20=b4=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >
Subgroups: 304 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C4⋊D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4⋊Dic5, C23.D5, C2×C4×D5, C2×C5⋊D4, D4×C10, C20⋊2D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C20⋊2D4
►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 72 27 51)(2 71 28 50)(3 70 29 49)(4 69 30 48)(5 68 31 47)(6 67 32 46)(7 66 33 45)(8 65 34 44)(9 64 35 43)(10 63 36 42)(11 62 37 41)(12 61 38 60)(13 80 39 59)(14 79 40 58)(15 78 21 57)(16 77 22 56)(17 76 23 55)(18 75 24 54)(19 74 25 53)(20 73 26 52)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)(41 72)(42 61)(43 70)(44 79)(45 68)(46 77)(47 66)(48 75)(49 64)(50 73)(51 62)(52 71)(53 80)(54 69)(55 78)(56 67)(57 76)(58 65)(59 74)(60 63)
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), (1,72,27,51)(2,71,28,50)(3,70,29,49)(4,69,30,48)(5,68,31,47)(6,67,32,46)(7,66,33,45)(8,65,34,44)(9,64,35,43)(10,63,36,42)(11,62,37,41)(12,61,38,60)(13,80,39,59)(14,79,40,58)(15,78,21,57)(16,77,22,56)(17,76,23,55)(18,75,24,54)(19,74,25,53)(20,73,26,52), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(41,72)(42,61)(43,70)(44,79)(45,68)(46,77)(47,66)(48,75)(49,64)(50,73)(51,62)(52,71)(53,80)(54,69)(55,78)(56,67)(57,76)(58,65)(59,74)(60,63)>;
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), (1,72,27,51)(2,71,28,50)(3,70,29,49)(4,69,30,48)(5,68,31,47)(6,67,32,46)(7,66,33,45)(8,65,34,44)(9,64,35,43)(10,63,36,42)(11,62,37,41)(12,61,38,60)(13,80,39,59)(14,79,40,58)(15,78,21,57)(16,77,22,56)(17,76,23,55)(18,75,24,54)(19,74,25,53)(20,73,26,52), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(41,72)(42,61)(43,70)(44,79)(45,68)(46,77)(47,66)(48,75)(49,64)(50,73)(51,62)(52,71)(53,80)(54,69)(55,78)(56,67)(57,76)(58,65)(59,74)(60,63) );
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)], [(1,72,27,51),(2,71,28,50),(3,70,29,49),(4,69,30,48),(5,68,31,47),(6,67,32,46),(7,66,33,45),(8,65,34,44),(9,64,35,43),(10,63,36,42),(11,62,37,41),(12,61,38,60),(13,80,39,59),(14,79,40,58),(15,78,21,57),(16,77,22,56),(17,76,23,55),(18,75,24,54),(19,74,25,53),(20,73,26,52)], [(1,11),(2,20),(3,9),(4,18),(5,7),(6,16),(8,14),(10,12),(13,19),(15,17),(21,23),(22,32),(24,30),(25,39),(26,28),(27,37),(29,35),(31,33),(34,40),(36,38),(41,72),(42,61),(43,70),(44,79),(45,68),(46,77),(47,66),(48,75),(49,64),(50,73),(51,62),(52,71),(53,80),(54,69),(55,78),(56,67),(57,76),(58,65),(59,74),(60,63)]])
C20⋊2D4 is a maximal subgroup of
D10.SD16 D10.12D8 D10⋊D8 D10.16SD16 D10⋊SD16 C40⋊6C4⋊C2 C5⋊2C8⋊D4 C5⋊(C8⋊2D4) C40⋊5C4⋊C2 D20⋊D4 C40⋊6D4 Dic10⋊D4 C40⋊12D4 D10⋊8SD16 C40⋊14D4 D20⋊7D4 C40⋊8D4 C42.228D10 D20⋊24D4 C42.229D10 C42.113D10 C42.115D10 C42.116D10 C42.117D10 C24⋊3D10 C24.33D10 C24.35D10 C24.36D10 D5×C4⋊D4 C4⋊C4⋊21D10 C10.382+ 1+4 C10.392+ 1+4 C10.732- 1+4 D20⋊20D4 C10.422+ 1+4 C10.432+ 1+4 C10.442+ 1+4 C10.452+ 1+4 C10.1152+ 1+4 C10.472+ 1+4 C10.482+ 1+4 C10.742- 1+4 C10.612+ 1+4 C10.632+ 1+4 C10.642+ 1+4 C10.692+ 1+4 D20⋊10D4 Dic10⋊10D4 C42.234D10 C42.144D10 C42.238D10 D20⋊11D4 Dic10⋊11D4 C42.168D10 D4×C5⋊D4 C24.41D10 C24.42D10 (C2×C20)⋊15D4 C10.1072- 1+4 C10.1472+ 1+4 C10.1482+ 1+4 C60⋊D4 C20⋊2D12 (C6×D5)⋊D4 D30⋊7D4 C60⋊2D4
C20⋊2D4 is a maximal quotient of
C24.3D10 C24.6D10 C24.12D10 C24.16D10 C20⋊4(C4⋊C4) (C2×C20).287D4 D10⋊4(C4⋊C4) (C2×C20).290D4 C42.61D10 D20.23D4 C20⋊2D8 Dic10⋊9D4 C20⋊5SD16 C20⋊Q16 C40⋊6D4 C40⋊12D4 C40.23D4 C40⋊14D4 C40⋊8D4 C40.44D4 D10⋊3Q16 C40.36D4 C40.29D4 C24.19D10 C24.20D10 C24.21D10 C60⋊D4 C20⋊2D12 (C6×D5)⋊D4 D30⋊7D4 C60⋊2D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 2 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | C20⋊2D4 | C4⋊Dic5 | C23.D5 | C2×C4×D5 | C2×C5⋊D4 | D4×C10 | C20 | D10 | C2×D4 | C10 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of C20⋊2D4 ►in GL4(𝔽41) generated by
| 1 | 1 | 0 | 0 |
| 5 | 6 | 0 | 0 |
| 0 | 0 | 8 | 14 |
| 0 | 0 | 10 | 33 |
| 20 | 3 | 0 | 0 |
| 3 | 21 | 0 | 0 |
| 0 | 0 | 1 | 23 |
| 0 | 0 | 0 | 40 |
| 6 | 7 | 0 | 0 |
| 36 | 35 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,5,0,0,1,6,0,0,0,0,8,10,0,0,14,33],[20,3,0,0,3,21,0,0,0,0,1,0,0,0,23,40],[6,36,0,0,7,35,0,0,0,0,40,0,0,0,0,40] >;
C20⋊2D4 in GAP, Magma, Sage, TeX
C_{20}\rtimes_2D_4 % in TeX
G:=Group("C20:2D4"); // GroupNames label
G:=SmallGroup(160,159);
// by ID
G=gap.SmallGroup(160,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations