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## G = D4×Dic5order 160 = 25·5

### Direct product of D4 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5
 Lower central C5 — C10 — D4×Dic5
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×Dic5
G = < a,b,c,d | a4=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 232 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C5, C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4, C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×4], C22×C10 [×2], C4×Dic5, C4⋊Dic5, C23.D5 [×2], C22×Dic5 [×2], D4×C10, D4×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×Dic5 [×6], C22×D5, D4×D5, D42D5, C22×Dic5, D4×Dic5

Smallest permutation representation of D4×Dic5
On 80 points
Generators in S80
(1 45 28 38)(2 46 29 39)(3 47 30 40)(4 48 21 31)(5 49 22 32)(6 50 23 33)(7 41 24 34)(8 42 25 35)(9 43 26 36)(10 44 27 37)(11 62 77 57)(12 63 78 58)(13 64 79 59)(14 65 80 60)(15 66 71 51)(16 67 72 52)(17 68 73 53)(18 69 74 54)(19 70 75 55)(20 61 76 56)
(1 38)(2 39)(3 40)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 61)(21 48)(22 49)(23 50)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 57 6 52)(2 56 7 51)(3 55 8 60)(4 54 9 59)(5 53 10 58)(11 50 16 45)(12 49 17 44)(13 48 18 43)(14 47 19 42)(15 46 20 41)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)

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

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

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

40 conjugacy classes

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

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 Dic5 D10 D4×D5 D4⋊2D5 kernel D4×Dic5 C4×Dic5 C4⋊Dic5 C23.D5 C22×Dic5 D4×C10 C5×D4 Dic5 C2×D4 C10 C2×C4 D4 C23 C2 C2 # reps 1 1 1 2 2 1 8 2 2 2 2 8 4 2 2

Matrix representation of D4×Dic5 in GL5(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 1 0 0 0 5 35
,
 32 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 6 0

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,5,0,0,0,1,35],[32,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,7,0] >;

D4×Dic5 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_5
% in TeX

G:=Group("D4xDic5");
// GroupNames label

G:=SmallGroup(160,155);
// by ID

G=gap.SmallGroup(160,155);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,4613]);
// Polycyclic

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

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