D4xDic5 - GroupNames

G = D₄×Dic₅ order 160 = 2⁵·5

Direct product of D₄ and Dic₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄×Dic₅, C₂³.₁₇D₁₀, C₅⋊₆(C₄×D₄), C₂₀⋊₆(C₂×C₄), (C₅×D₄)⋊₆C₄, C₂.₅(D₄×D₅), (C₂×D₄).₇D₅, C₄⋊₁(C₂×Dic₅), C₄⋊Dic₅⋊₁₃C₂, (C₄×Dic₅)⋊₄C₂, (D₄×C₁₀).₄C₂, (C₂×C₄).₄₉D₁₀, C₁₀.₃₇(C₂×D₄), C₂³.D₅⋊₇C₂, C₂²⋊₁(C₂×Dic₅), C₁₀.₂₈(C₄○D₄), C₂.₅(D₄⋊₂D₅), (C₂×C₂₀).₃₂C₂², C₁₀.₃₈(C₂²×C₄), (C₂×C₁₀).₄₉C₂³, (C₂²×Dic₅)⋊₄C₂, C₂.₆(C₂²×Dic₅), C₂².₂₅(C₂²×D₅), (C₂²×C₁₀).₁₇C₂², (C₂×Dic₅).₄₀C₂², (C₂×C₁₀)⋊₆(C₂×C₄), SmallGroup(160,155)

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, C₂, C₄, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₄×D₄, C₂×Dic₅, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₄×Dic₅, C₄⋊Dic₅, C₂³.D₅, C₂²×Dic₅, D₄×C₁₀, D₄×Dic₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₅, C₂²×C₄, C₂×D₄, C₄○D₄, Dic₅, D₁₀, C₄×D₄, C₂×Dic₅, C₂²×D₅, D₄×D₅, D₄⋊₂D₅, 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)]])

D₄×Dic₅ is a maximal subgroup of
Dic₅.₂₃D₈ Dic₅⋊₄D₈ D₄.D₅⋊₅C₄ Dic₅⋊₆SD₁₆ Dic₅.₁₄D₈ D₄⋊Dic₁₀ D₄.Dic₁₀ D₄.₂Dic₁₀ D₄⋊D₅⋊₆C₄ Dic₅⋊D₈ D₈⋊Dic₅ (C₂×D₈).D₅ Dic₅⋊₃SD₁₆ SD₁₆⋊Dic₅ (C₅×D₄).D₄ C₅⋊C₈⋊₇D₄ C₂₀⋊₂M₄(2) C₄×D₄⋊₂D₅ D₄⋊₅Dic₁₀ D₄⋊₆Dic₁₀ C₄×D₄×D₅ C₄²⋊₁₁D₁₀ C₄².₁₀₈D₁₀ C₂⁴.₅₆D₁₀ C₂⁴.₃₂D₁₀ C₂⁴.₃₃D₁₀ C₂⁴.₃₅D₁₀ C₂₀⋊(C₄○D₄) Dic₁₀⋊₁₉D₄ C₄⋊C₄.₁₇₈D₁₀ C₁₀.₃₄2₊¹⁺⁴ C₁₀.₃₅2₊¹⁺⁴ C₁₀.₃₆2₊¹⁺⁴ C₄⋊C₄⋊₂₁D₁₀ C₁₀.₇₃2_-¹⁺⁴ C₁₀.₄₃2₊¹⁺⁴ C₁₀.₄₅2₊¹⁺⁴ C₁₀.₄₆2₊¹⁺⁴ C₁₀.₁₁₅2₊¹⁺⁴ C₁₀.₄₇2₊¹⁺⁴ C₄⋊C₄.₁₉₇D₁₀ C₁₀.₈₀2_-¹⁺⁴ C₁₀.₁₂₂2₊¹⁺⁴ C₁₀.₈₅2_-¹⁺⁴ C₄².₁₃₉D₁₀ C₄².₂₃₄D₁₀ C₄².₁₄₃D₁₀ C₄².₁₄₄D₁₀ C₄².₁₆₆D₁₀ C₄².₂₃₈D₁₀ Dic₁₀⋊₁₁D₄ C₄².₁₆₈D₁₀ C₂⁴.₃₈D₁₀ C₂⁴.₄₂D₁₀ C₁₀.₁₀₄2_-¹⁺⁴ C₁₀.₁₀₆2_-¹⁺⁴ C₁₀.₁₄₅2₊¹⁺⁴ Dic₁₅⋊₈D₄ Dic₁₅⋊₁₇D₄
D₄×Dic₅ is a maximal quotient of
C₂⁴.₄₇D₁₀ C₂⁴.₈D₁₀ C₄⋊C₄⋊₅Dic₅ C₂₀⋊₆(C₄⋊C₄) C₄².₄₇D₁₀ C₂₀⋊₇M₄(2) D₈⋊Dic₅ SD₁₆⋊Dic₅ Q₁₆⋊Dic₅ D₈⋊₅Dic₅ D₈⋊₄Dic₅ C₂⁴.₁₈D₁₀ C₂⁴.₁₉D₁₀ Dic₁₅⋊₈D₄ Dic₁₅⋊₁₇D₄

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

G = D4×Dic5 order 160 = 25·5

Direct product of D4 and Dic5

G = D₄×Dic₅ order 160 = 2⁵·5

Direct product of D₄ and Dic₅