metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊6C4, C4.9D20, C10.7D8, C20.1D4, C10.7SD16, C4⋊C4⋊1D5, C4.1(C4×D5), C5⋊2(D4⋊C4), C20.24(C2×C4), C2.2(D4⋊D5), (C2×D20).6C2, (C2×C4).35D10, (C2×C10).30D4, C2.2(Q8⋊D5), (C2×C20).10C22, C2.5(D10⋊C4), C10.14(C22⋊C4), C22.14(C5⋊D4), (C5×C4⋊C4)⋊1C2, (C2×C5⋊2C8)⋊1C2, SmallGroup(160,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊6C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a15b >
Smallest permutation representation of D20⋊6C4
►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 23)(2 22)(3 21)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 64)(42 63)(43 62)(44 61)(45 80)(46 79)(47 78)(48 77)(49 76)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)
(1 66 24 55)(2 77 25 46)(3 68 26 57)(4 79 27 48)(5 70 28 59)(6 61 29 50)(7 72 30 41)(8 63 31 52)(9 74 32 43)(10 65 33 54)(11 76 34 45)(12 67 35 56)(13 78 36 47)(14 69 37 58)(15 80 38 49)(16 71 39 60)(17 62 40 51)(18 73 21 42)(19 64 22 53)(20 75 23 44)
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), (1,23)(2,22)(3,21)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,64)(42,63)(43,62)(44,61)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65), (1,66,24,55)(2,77,25,46)(3,68,26,57)(4,79,27,48)(5,70,28,59)(6,61,29,50)(7,72,30,41)(8,63,31,52)(9,74,32,43)(10,65,33,54)(11,76,34,45)(12,67,35,56)(13,78,36,47)(14,69,37,58)(15,80,38,49)(16,71,39,60)(17,62,40,51)(18,73,21,42)(19,64,22,53)(20,75,23,44)>;
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), (1,23)(2,22)(3,21)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(41,64)(42,63)(43,62)(44,61)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65), (1,66,24,55)(2,77,25,46)(3,68,26,57)(4,79,27,48)(5,70,28,59)(6,61,29,50)(7,72,30,41)(8,63,31,52)(9,74,32,43)(10,65,33,54)(11,76,34,45)(12,67,35,56)(13,78,36,47)(14,69,37,58)(15,80,38,49)(16,71,39,60)(17,62,40,51)(18,73,21,42)(19,64,22,53)(20,75,23,44) );
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)], [(1,23),(2,22),(3,21),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(41,64),(42,63),(43,62),(44,61),(45,80),(46,79),(47,78),(48,77),(49,76),(50,75),(51,74),(52,73),(53,72),(54,71),(55,70),(56,69),(57,68),(58,67),(59,66),(60,65)], [(1,66,24,55),(2,77,25,46),(3,68,26,57),(4,79,27,48),(5,70,28,59),(6,61,29,50),(7,72,30,41),(8,63,31,52),(9,74,32,43),(10,65,33,54),(11,76,34,45),(12,67,35,56),(13,78,36,47),(14,69,37,58),(15,80,38,49),(16,71,39,60),(17,62,40,51),(18,73,21,42),(19,64,22,53),(20,75,23,44)]])
D20⋊6C4 is a maximal subgroup of
Dic5.5D8 Dic10⋊2D4 C4⋊C4.D10 D5×D4⋊C4 (D4×D5)⋊C4 D4⋊D20 D20.8D4 D20⋊3D4 Q8⋊C4⋊D5 Dic10.11D4 Q8⋊Dic5⋊C2 Q8⋊(C4×D5) Q8⋊2D5⋊C4 Q8.D20 D20⋊4D4 D20.12D4 Dic5⋊8SD16 D10.17SD16 C8⋊8D20 C8⋊2D20 C4.Q8⋊D5 D40⋊15C4 D20⋊Q8 D20.Q8 D40⋊12C4 D10.13D8 C8⋊7D20 C2.D8⋊D5 C8⋊3D20 C40⋊21(C2×C4) D20⋊2Q8 D20.2Q8 C4○D20⋊9C4 (C2×C10).40D8 C4⋊C4.228D10 C4⋊C4⋊36D10 C4○D20⋊10C4 C4⋊C4.236D10 C4×D4⋊D5 C42.48D10 C20⋊7D8 D4.1D20 C4×Q8⋊D5 C42.56D10 Q8⋊D20 Q8.1D20 D20⋊16D4 D20⋊17D4 (C2×C10)⋊D8 C4⋊D4⋊D5 D20.36D4 D20.37D4 C5⋊2C8⋊24D4 C22⋊Q8⋊D5 D20.4Q8 C42.70D10 C42.216D10 D20⋊5Q8 D20⋊6Q8 C20.D8 C42.82D10 C30.D8 D60⋊15C4 D60⋊9C4
D20⋊6C4 is a maximal quotient of
(C2×D20)⋊C4 C20.47D8 D20⋊4C8 C4.D40 D40⋊14C4 C40.5D4 C10.Q32 D40.6C4 C40.8D4 D40.5C4 C20.31C42 C30.D8 D60⋊15C4 D60⋊9C4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4×D5 | D20 | C5⋊D4 | D4⋊D5 | Q8⋊D5 |
| kernel | D20⋊6C4 | C2×C5⋊2C8 | C5×C4⋊C4 | C2×D20 | D20 | C20 | C2×C10 | C4⋊C4 | C10 | C10 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of D20⋊6C4 ►in GL4(𝔽41) generated by
| 35 | 1 | 0 | 0 |
| 5 | 40 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 30 | 11 |
| 0 | 0 | 15 | 11 |
G:=sub<GL(4,GF(41))| [35,5,0,0,1,40,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,1,40,0,0,0,0,40,0,0,0,2,1],[9,0,0,0,0,9,0,0,0,0,30,15,0,0,11,11] >;
D20⋊6C4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_6C_4 % in TeX
G:=Group("D20:6C4"); // GroupNames label
G:=SmallGroup(160,16);
// by ID
G=gap.SmallGroup(160,16);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,579,297,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations
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