metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊8C4, Dic5⋊5D4, C5⋊4(C4×D4), C4⋊C4⋊8D5, C4⋊1(C4×D5), C20⋊5(C2×C4), C2.4(D4×D5), D10⋊4(C2×C4), (C4×Dic5)⋊3C2, (C2×D20).8C2, (C2×C4).31D10, C10.24(C2×D4), D10⋊C4⋊12C2, C10.33(C4○D4), (C2×C10).34C23, (C2×C20).24C22, C10.24(C22×C4), C2.2(Q8⋊2D5), C22.18(C22×D5), (C2×Dic5).63C22, (C22×D5).23C22, (C5×C4⋊C4)⋊4C2, (C2×C4×D5)⋊12C2, C2.13(C2×C4×D5), SmallGroup(160,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊8C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >
Subgroups: 312 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C5, C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5, C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, D20⋊8C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C2×C4×D5, D4×D5, Q8⋊2D5, D20⋊8C4
►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 20)(17 19)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
(1 64 41 21)(2 75 42 32)(3 66 43 23)(4 77 44 34)(5 68 45 25)(6 79 46 36)(7 70 47 27)(8 61 48 38)(9 72 49 29)(10 63 50 40)(11 74 51 31)(12 65 52 22)(13 76 53 33)(14 67 54 24)(15 78 55 35)(16 69 56 26)(17 80 57 37)(18 71 58 28)(19 62 59 39)(20 73 60 30)
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)(16,20)(17,19)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,64,41,21)(2,75,42,32)(3,66,43,23)(4,77,44,34)(5,68,45,25)(6,79,46,36)(7,70,47,27)(8,61,48,38)(9,72,49,29)(10,63,50,40)(11,74,51,31)(12,65,52,22)(13,76,53,33)(14,67,54,24)(15,78,55,35)(16,69,56,26)(17,80,57,37)(18,71,58,28)(19,62,59,39)(20,73,60,30)>;
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)(16,20)(17,19)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(56,60)(57,59)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,64,41,21)(2,75,42,32)(3,66,43,23)(4,77,44,34)(5,68,45,25)(6,79,46,36)(7,70,47,27)(8,61,48,38)(9,72,49,29)(10,63,50,40)(11,74,51,31)(12,65,52,22)(13,76,53,33)(14,67,54,24)(15,78,55,35)(16,69,56,26)(17,80,57,37)(18,71,58,28)(19,62,59,39)(20,73,60,30) );
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),(16,20),(17,19),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(56,60),(57,59),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)], [(1,64,41,21),(2,75,42,32),(3,66,43,23),(4,77,44,34),(5,68,45,25),(6,79,46,36),(7,70,47,27),(8,61,48,38),(9,72,49,29),(10,63,50,40),(11,74,51,31),(12,65,52,22),(13,76,53,33),(14,67,54,24),(15,78,55,35),(16,69,56,26),(17,80,57,37),(18,71,58,28),(19,62,59,39),(20,73,60,30)])
D20⋊8C4 is a maximal subgroup of
D20⋊C8 Dic5⋊4D8 D4⋊D5⋊6C4 D20⋊3D4 D20.D4 Dic5⋊7SD16 Q8⋊D5⋊6C4 Dic5⋊SD16 D20.12D4 Dic5⋊8SD16 D40⋊15C4 D20⋊Q8 D20.Q8 D40⋊12C4 C40⋊21(C2×C4) D20⋊2Q8 D20.2Q8 D20⋊2C8 D10⋊2M4(2) C20⋊M4(2) C10.82+ 1+4 C10.2- 1+4 C10.112+ 1+4 C42⋊7D10 C42.188D10 C42.95D10 C42.97D10 C4×D4×D5 C42⋊11D10 Dic10⋊24D4 C42.114D10 C42.122D10 C4×Q8⋊2D5 C42.126D10 C42.136D10 C20⋊(C4○D4) D20⋊19D4 D20⋊20D4 C10.472+ 1+4 C22⋊Q8⋊25D5 C4⋊C4⋊26D10 D20⋊21D4 D20⋊22D4 Dic10⋊22D4 C10.532+ 1+4 C10.202- 1+4 C10.242- 1+4 C10.1212+ 1+4 C4⋊C4⋊28D10 C10.612+ 1+4 C10.642+ 1+4 D20⋊7Q8 C42.237D10 C42.150D10 C42.151D10 C42.153D10 C42.156D10 C42⋊23D10 C42⋊24D10 C42.189D10 C42.163D10 C42.240D10 D20⋊8Q8 D20⋊9Q8 C42.178D10 Dic15⋊13D4 D60⋊17C4 D20⋊8Dic3 C15⋊20(C4×D4) D60⋊11C4
D20⋊8C4 is a maximal quotient of
C4⋊Dic5⋊15C4 (C2×C4)⋊9D20 D10⋊2C42 C10.54(C4×D4) D20⋊5C8 D10⋊5M4(2) C20⋊6M4(2) Dic5⋊8SD16 Dic20⋊15C4 D40⋊15C4 D40⋊12C4 Dic5⋊5Q16 C40⋊21(C2×C4) D40⋊16C4 D40⋊13C4 C20⋊4(C4⋊C4) C4⋊C4×Dic5 (C2×D20)⋊22C4 D10⋊5(C4⋊C4) C10.90(C4×D4) Dic15⋊13D4 D60⋊17C4 D20⋊8Dic3 C15⋊20(C4×D4) D60⋊11C4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D4×D5 | Q8⋊2D5 |
| kernel | D20⋊8C4 | C4×Dic5 | D10⋊C4 | C5×C4⋊C4 | C2×C4×D5 | C2×D20 | D20 | Dic5 | C4⋊C4 | C10 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of D20⋊8C4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 6 | 40 | 0 | 0 |
| 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 21 |
| 0 | 0 | 0 | 37 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 37 | 1 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 16 |
| 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,6,36,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,21,1],[40,0,0,0,0,0,40,0,0,0,0,40,1,0,0,0,0,0,40,37,0,0,0,0,1],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,16,32] >;
D20⋊8C4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_8C_4 % in TeX
G:=Group("D20:8C4"); // GroupNames label
G:=SmallGroup(160,114);
// by ID
G=gap.SmallGroup(160,114);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,188,50,4613]);
// 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^10*b>;
// generators/relations