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## G = D20⋊11D4order 320 = 26·5

### 4th semidirect product of D20 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊11D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — D20⋊11D4
 Lower central C5 — C2×C10 — D20⋊11D4
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for D2011D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, bd=db, dcd=c-1 >

Subgroups: 1974 in 428 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×44], C5, C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×34], C23 [×4], C23 [×24], D5 [×8], C10, C10 [×2], C10 [×4], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×6], C2×D4 [×26], C24 [×4], Dic5 [×4], C20 [×4], C20, D10 [×8], D10 [×24], C2×C10, C2×C10 [×12], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C41D4, C22×D4 [×4], C4×D5 [×8], D20 [×8], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20, C2×C20 [×2], C5×D4 [×10], C22×D5 [×4], C22×D5 [×20], C22×C10 [×4], D42, C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C2×C4×D5 [×4], C2×D20 [×2], D4×D5 [×16], C2×C5⋊D4 [×8], D4×C10 [×6], C23×D5 [×4], C4×D20 [×2], C23⋊D10 [×4], C202D4 [×4], C5×C41D4, C2×D4×D5 [×4], D2011D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ 1+4, C22×D5 [×7], D42, D4×D5 [×4], C23×D5, C2×D4×D5 [×2], D46D10, D2011D4

Smallest permutation representation of D2011D4
On 80 points
Generators in S80
```(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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)
(1 40 68 52)(2 21 69 53)(3 22 70 54)(4 23 71 55)(5 24 72 56)(6 25 73 57)(7 26 74 58)(8 27 75 59)(9 28 76 60)(10 29 77 41)(11 30 78 42)(12 31 79 43)(13 32 80 44)(14 33 61 45)(15 34 62 46)(16 35 63 47)(17 36 64 48)(18 37 65 49)(19 38 66 50)(20 39 67 51)
(1 73)(2 64)(3 75)(4 66)(5 77)(6 68)(7 79)(8 70)(9 61)(10 72)(11 63)(12 74)(13 65)(14 76)(15 67)(16 78)(17 69)(18 80)(19 71)(20 62)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)(41 56)(42 47)(43 58)(44 49)(45 60)(46 51)(48 53)(50 55)(52 57)(54 59)```

`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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (1,40,68,52)(2,21,69,53)(3,22,70,54)(4,23,71,55)(5,24,72,56)(6,25,73,57)(7,26,74,58)(8,27,75,59)(9,28,76,60)(10,29,77,41)(11,30,78,42)(12,31,79,43)(13,32,80,44)(14,33,61,45)(15,34,62,46)(16,35,63,47)(17,36,64,48)(18,37,65,49)(19,38,66,50)(20,39,67,51), (1,73)(2,64)(3,75)(4,66)(5,77)(6,68)(7,79)(8,70)(9,61)(10,72)(11,63)(12,74)(13,65)(14,76)(15,67)(16,78)(17,69)(18,80)(19,71)(20,62)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,56)(42,47)(43,58)(44,49)(45,60)(46,51)(48,53)(50,55)(52,57)(54,59)>;`

`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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (1,40,68,52)(2,21,69,53)(3,22,70,54)(4,23,71,55)(5,24,72,56)(6,25,73,57)(7,26,74,58)(8,27,75,59)(9,28,76,60)(10,29,77,41)(11,30,78,42)(12,31,79,43)(13,32,80,44)(14,33,61,45)(15,34,62,46)(16,35,63,47)(17,36,64,48)(18,37,65,49)(19,38,66,50)(20,39,67,51), (1,73)(2,64)(3,75)(4,66)(5,77)(6,68)(7,79)(8,70)(9,61)(10,72)(11,63)(12,74)(13,65)(14,76)(15,67)(16,78)(17,69)(18,80)(19,71)(20,62)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(41,56)(42,47)(43,58)(44,49)(45,60)(46,51)(48,53)(50,55)(52,57)(54,59) );`

`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)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,80),(9,79),(10,78),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,70),(19,69),(20,68),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41)], [(1,40,68,52),(2,21,69,53),(3,22,70,54),(4,23,71,55),(5,24,72,56),(6,25,73,57),(7,26,74,58),(8,27,75,59),(9,28,76,60),(10,29,77,41),(11,30,78,42),(12,31,79,43),(13,32,80,44),(14,33,61,45),(15,34,62,46),(16,35,63,47),(17,36,64,48),(18,37,65,49),(19,38,66,50),(20,39,67,51)], [(1,73),(2,64),(3,75),(4,66),(5,77),(6,68),(7,79),(8,70),(9,61),(10,72),(11,63),(12,74),(13,65),(14,76),(15,67),(16,78),(17,69),(18,80),(19,71),(20,62),(21,36),(22,27),(23,38),(24,29),(25,40),(26,31),(28,33),(30,35),(32,37),(34,39),(41,56),(42,47),(43,58),(44,49),(45,60),(46,51),(48,53),(50,55),(52,57),(54,59)])`

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H ··· 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 2 2 2 ··· 2 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 4 4 4 4 10 ··· 10 2 2 2 2 4 20 20 20 20 2 2 2 ··· 2 8 ··· 8 4 ··· 4

53 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 D10 D10 2+ 1+4 D4×D5 D4⋊6D10 kernel D20⋊11D4 C4×D20 C23⋊D10 C20⋊2D4 C5×C4⋊1D4 C2×D4×D5 D20 C4⋊1D4 C42 C2×D4 C10 C4 C2 # reps 1 2 4 4 1 4 8 2 2 12 1 8 4

Matrix representation of D2011D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 6 0 0 0 0 0 0 40 39 0 0 0 0 1 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 2 0 0 0 0 0 40
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 39 0 0 0 0 1 1
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 39 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1] >;`

D2011D4 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_{11}D_4`
`% in TeX`

`G:=Group("D20:11D4");`
`// GroupNames label`

`G:=SmallGroup(320,1389);`
`// by ID`

`G=gap.SmallGroup(320,1389);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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