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G = D10.13D4order 160 = 25·5

2nd non-split extension by D10 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10.13D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.13D4
 Lower central C5 — C2×C10 — D10.13D4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for D10.13D4
G = < a,b,c,d | a10=b2=c4=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 288 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22, C22 [×7], C5, C2×C4 [×3], C2×C4 [×4], D4 [×2], C23 [×2], D5 [×3], C10 [×3], C22⋊C4 [×3], C4⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C22.D4, C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C10.D4, D10⋊C4 [×3], C5×C4⋊C4, C2×C4×D5, C2×D20, D10.13D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C22×D5, C4○D20, D4×D5, Q82D5, D10.13D4

Smallest permutation representation of D10.13D4
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 15)(2 14)(3 13)(4 12)(5 11)(6 20)(7 19)(8 18)(9 17)(10 16)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 80)(30 79)(31 63)(32 62)(33 61)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(41 53)(42 52)(43 51)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)
(1 50 30 40)(2 41 21 31)(3 42 22 32)(4 43 23 33)(5 44 24 34)(6 45 25 35)(7 46 26 36)(8 47 27 37)(9 48 28 38)(10 49 29 39)(11 55 75 65)(12 56 76 66)(13 57 77 67)(14 58 78 68)(15 59 79 69)(16 60 80 70)(17 51 71 61)(18 52 72 62)(19 53 73 63)(20 54 74 64)
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 56 36 51)(32 57 37 52)(33 58 38 53)(34 59 39 54)(35 60 40 55)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 10A ··· 10F 20A ··· 20L order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 20 2 2 4 4 10 10 20 2 2 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 C4○D20 D4×D5 Q8⋊2D5 kernel D10.13D4 C10.D4 D10⋊C4 C5×C4⋊C4 C2×C4×D5 C2×D20 D10 C4⋊C4 C10 C2×C4 C2 C2 C2 # reps 1 1 3 1 1 1 2 2 4 6 8 2 2

Matrix representation of D10.13D4 in GL4(𝔽41) generated by

 0 6 0 0 34 7 0 0 0 0 40 0 0 0 0 40
,
 27 16 0 0 16 14 0 0 0 0 0 9 0 0 32 0
,
 24 35 0 0 7 17 0 0 0 0 9 0 0 0 0 32
,
 30 13 0 0 19 11 0 0 0 0 0 32 0 0 32 0
`G:=sub<GL(4,GF(41))| [0,34,0,0,6,7,0,0,0,0,40,0,0,0,0,40],[27,16,0,0,16,14,0,0,0,0,0,32,0,0,9,0],[24,7,0,0,35,17,0,0,0,0,9,0,0,0,0,32],[30,19,0,0,13,11,0,0,0,0,0,32,0,0,32,0] >;`

D10.13D4 in GAP, Magma, Sage, TeX

`D_{10}._{13}D_4`
`% in TeX`

`G:=Group("D10.13D4");`
`// GroupNames label`

`G:=SmallGroup(160,115);`
`// by ID`

`G=gap.SmallGroup(160,115);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,55,218,188,86,4613]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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