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## G = D20.29D4order 320 = 26·5

### 12nd non-split extension by D20 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20.29D4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — D4⋊6D10 — D20.29D4
 Lower central C5 — C10 — C20 — D20.29D4
 Upper central C1 — C2 — C2×C4 — C2×SD16

Generators and relations for D20.29D4
G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c3 >

Subgroups: 1014 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×14], Q8 [×2], Q8 [×6], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8 [×3], SD16 [×4], SD16 [×6], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8 [×3], C4○D4 [×11], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×4], C2×C10, C2×C10 [×3], C8○D4, C2×SD16, C2×SD16 [×2], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, C52C8 [×2], C40 [×2], Dic10 [×3], Dic10 [×2], C4×D5 [×2], C4×D5 [×6], D20 [×3], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×6], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×D5 [×2], C22×C10, D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×SD16 [×4], C4○D20 [×3], C4○D20 [×2], D4×D5 [×2], D4×D5, D42D5 [×2], D42D5, Q8×D5 [×2], Q8×D5, Q82D5 [×2], Q82D5, C2×C5⋊D4 [×2], D4×C10, Q8×C10, D20.3C4, D407C2, D5×SD16 [×2], D40⋊C2 [×2], SD16⋊D5 [×2], SD163D5 [×2], D4.D10, C20.C23, C10×SD16, D46D10, Q8.10D10, D20.29D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○SD16, D4×D5 [×2], C23×D5, C2×D4×D5, D20.29D4

Smallest permutation representation of D20.29D4
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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)(41 43)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)
(1 50 63 37 11 60 73 27)(2 51 64 38 12 41 74 28)(3 52 65 39 13 42 75 29)(4 53 66 40 14 43 76 30)(5 54 67 21 15 44 77 31)(6 55 68 22 16 45 78 32)(7 56 69 23 17 46 79 33)(8 57 70 24 18 47 80 34)(9 58 71 25 19 48 61 35)(10 59 72 26 20 49 62 36)
(1 78)(2 69)(3 80)(4 71)(5 62)(6 73)(7 64)(8 75)(9 66)(10 77)(11 68)(12 79)(13 70)(14 61)(15 72)(16 63)(17 74)(18 65)(19 76)(20 67)(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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,43)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76), (1,50,63,37,11,60,73,27)(2,51,64,38,12,41,74,28)(3,52,65,39,13,42,75,29)(4,53,66,40,14,43,76,30)(5,54,67,21,15,44,77,31)(6,55,68,22,16,45,78,32)(7,56,69,23,17,46,79,33)(8,57,70,24,18,47,80,34)(9,58,71,25,19,48,61,35)(10,59,72,26,20,49,62,36), (1,78)(2,69)(3,80)(4,71)(5,62)(6,73)(7,64)(8,75)(9,66)(10,77)(11,68)(12,79)(13,70)(14,61)(15,72)(16,63)(17,74)(18,65)(19,76)(20,67)(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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,43)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76), (1,50,63,37,11,60,73,27)(2,51,64,38,12,41,74,28)(3,52,65,39,13,42,75,29)(4,53,66,40,14,43,76,30)(5,54,67,21,15,44,77,31)(6,55,68,22,16,45,78,32)(7,56,69,23,17,46,79,33)(8,57,70,24,18,47,80,34)(9,58,71,25,19,48,61,35)(10,59,72,26,20,49,62,36), (1,78)(2,69)(3,80)(4,71)(5,62)(6,73)(7,64)(8,75)(9,66)(10,77)(11,68)(12,79)(13,70)(14,61)(15,72)(16,63)(17,74)(18,65)(19,76)(20,67)(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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(38,40),(41,43),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(61,69),(62,68),(63,67),(64,66),(70,80),(71,79),(72,78),(73,77),(74,76)], [(1,50,63,37,11,60,73,27),(2,51,64,38,12,41,74,28),(3,52,65,39,13,42,75,29),(4,53,66,40,14,43,76,30),(5,54,67,21,15,44,77,31),(6,55,68,22,16,45,78,32),(7,56,69,23,17,46,79,33),(8,57,70,24,18,47,80,34),(9,58,71,25,19,48,61,35),(10,59,72,26,20,49,62,36)], [(1,78),(2,69),(3,80),(4,71),(5,62),(6,73),(7,64),(8,75),(9,66),(10,77),(11,68),(12,79),(13,70),(14,61),(15,72),(16,63),(17,74),(18,65),(19,76),(20,67),(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)])`

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 10 10 20 20 2 2 4 4 10 10 20 20 2 2 2 2 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D4○SD16 D4×D5 D4×D5 D20.29D4 kernel D20.29D4 D20.3C4 D40⋊7C2 D5×SD16 D40⋊C2 SD16⋊D5 SD16⋊3D5 D4.D10 C20.C23 C10×SD16 D4⋊6D10 Q8.10D10 Dic10 D20 C5⋊D4 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 2 2 8 2 2 2 2 2 8

Matrix representation of D20.29D4 in GL4(𝔽41) generated by

 0 0 1 24 0 0 17 38 40 17 0 0 24 3 0 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 26 0 15 0 0 26 0 15 26 0 26 0 0 26 0 26
,
 17 40 0 0 1 24 0 0 0 0 24 1 0 0 40 17
`G:=sub<GL(4,GF(41))| [0,0,40,24,0,0,17,3,1,17,0,0,24,38,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[26,0,26,0,0,26,0,26,15,0,26,0,0,15,0,26],[17,1,0,0,40,24,0,0,0,0,24,40,0,0,1,17] >;`

D20.29D4 in GAP, Magma, Sage, TeX

`D_{20}._{29}D_4`
`% in TeX`

`G:=Group("D20.29D4");`
`// GroupNames label`

`G:=SmallGroup(320,1434);`
`// by ID`

`G=gap.SmallGroup(320,1434);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,185,136,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^4=a^10,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^3>;`
`// generators/relations`

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