Q8xDic5 - GroupNames

G = Q₈×Dic₅ order 160 = 2⁵·5

Direct product of Q₈ and Dic₅

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

Aliases: Q₈×Dic₅, C₅⋊₄(C₄×Q₈), (C₅×Q₈)⋊₆C₄, C₂.₃(Q₈×D₅), (C₂×Q₈).₅D₅, C₂₀.₃₅(C₂×C₄), (C₂×C₄).₅₆D₁₀, (Q₈×C₁₀).₅C₂, C₁₀.₁₆(C₂×Q₈), C₄.₄(C₂×Dic₅), C₄⋊Dic₅.₁₂C₂, (C₄×Dic₅).₄C₂, C₁₀.₃₅(C₄○D₄), (C₂×C₂₀).₃₉C₂², C₁₀.₃₉(C₂²×C₄), (C₂×C₁₀).₅₇C₂³, C₂.₃(Q₈⋊₂D₅), C₂.₇(C₂²×Dic₅), C₂².₂₆(C₂²×D₅), (C₂×Dic₅).₄₃C₂², SmallGroup(160,166)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — Q₈×Dic₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₄×Dic₅ — Q₈×Dic₅

Lower central

C₅ — C₁₀ — Q₈×Dic₅

Upper central

C₁ — C₂² — C₂×Q₈

Generators and relations for Q₈×Dic₅
G = < a,b,c,d | a⁴=c¹⁰=1, b²=a², d²=c⁵, bab^-1=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 152 in 70 conjugacy classes, 51 normal (14 characteristic)
C₁, C₂, C₄, C₄, C₂², C₅, C₂×C₄, C₂×C₄, Q₈, C₁₀, C₄², C₄⋊C₄, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂×C₁₀, C₄×Q₈, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₅×Q₈, C₄×Dic₅, C₄⋊Dic₅, Q₈×C₁₀, Q₈×Dic₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, Q₈, C₂³, D₅, C₂²×C₄, C₂×Q₈, C₄○D₄, Dic₅, D₁₀, C₄×Q₈, C₂×Dic₅, C₂²×D₅, Q₈×D₅, Q₈⋊₂D₅, C₂²×Dic₅, Q₈×Dic₅

Smallest permutation representation of Q₈×Dic₅
►Regular action on 160 points

Generators in S₁₆₀

(1 48 28 36)(2 49 29 37)(3 50 30 38)(4 41 21 39)(5 42 22 40)(6 43 23 31)(7 44 24 32)(8 45 25 33)(9 46 26 34)(10 47 27 35)(11 139 159 142)(12 140 160 143)(13 131 151 144)(14 132 152 145)(15 133 153 146)(16 134 154 147)(17 135 155 148)(18 136 156 149)(19 137 157 150)(20 138 158 141)(51 71 63 90)(52 72 64 81)(53 73 65 82)(54 74 66 83)(55 75 67 84)(56 76 68 85)(57 77 69 86)(58 78 70 87)(59 79 61 88)(60 80 62 89)(91 124 104 111)(92 125 105 112)(93 126 106 113)(94 127 107 114)(95 128 108 115)(96 129 109 116)(97 130 110 117)(98 121 101 118)(99 122 102 119)(100 123 103 120)

(1 68 28 56)(2 69 29 57)(3 70 30 58)(4 61 21 59)(5 62 22 60)(6 63 23 51)(7 64 24 52)(8 65 25 53)(9 66 26 54)(10 67 27 55)(11 122 159 119)(12 123 160 120)(13 124 151 111)(14 125 152 112)(15 126 153 113)(16 127 154 114)(17 128 155 115)(18 129 156 116)(19 130 157 117)(20 121 158 118)(31 90 43 71)(32 81 44 72)(33 82 45 73)(34 83 46 74)(35 84 47 75)(36 85 48 76)(37 86 49 77)(38 87 50 78)(39 88 41 79)(40 89 42 80)(91 144 104 131)(92 145 105 132)(93 146 106 133)(94 147 107 134)(95 148 108 135)(96 149 109 136)(97 150 110 137)(98 141 101 138)(99 142 102 139)(100 143 103 140)

(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)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)

(1 99 6 94)(2 98 7 93)(3 97 8 92)(4 96 9 91)(5 95 10 100)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

G:=sub<Sym(160)| (1,48,28,36)(2,49,29,37)(3,50,30,38)(4,41,21,39)(5,42,22,40)(6,43,23,31)(7,44,24,32)(8,45,25,33)(9,46,26,34)(10,47,27,35)(11,139,159,142)(12,140,160,143)(13,131,151,144)(14,132,152,145)(15,133,153,146)(16,134,154,147)(17,135,155,148)(18,136,156,149)(19,137,157,150)(20,138,158,141)(51,71,63,90)(52,72,64,81)(53,73,65,82)(54,74,66,83)(55,75,67,84)(56,76,68,85)(57,77,69,86)(58,78,70,87)(59,79,61,88)(60,80,62,89)(91,124,104,111)(92,125,105,112)(93,126,106,113)(94,127,107,114)(95,128,108,115)(96,129,109,116)(97,130,110,117)(98,121,101,118)(99,122,102,119)(100,123,103,120), (1,68,28,56)(2,69,29,57)(3,70,30,58)(4,61,21,59)(5,62,22,60)(6,63,23,51)(7,64,24,52)(8,65,25,53)(9,66,26,54)(10,67,27,55)(11,122,159,119)(12,123,160,120)(13,124,151,111)(14,125,152,112)(15,126,153,113)(16,127,154,114)(17,128,155,115)(18,129,156,116)(19,130,157,117)(20,121,158,118)(31,90,43,71)(32,81,44,72)(33,82,45,73)(34,83,46,74)(35,84,47,75)(36,85,48,76)(37,86,49,77)(38,87,50,78)(39,88,41,79)(40,89,42,80)(91,144,104,131)(92,145,105,132)(93,146,106,133)(94,147,107,134)(95,148,108,135)(96,149,109,136)(97,150,110,137)(98,141,101,138)(99,142,102,139)(100,143,103,140), (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155)>;

G:=Group( (1,48,28,36)(2,49,29,37)(3,50,30,38)(4,41,21,39)(5,42,22,40)(6,43,23,31)(7,44,24,32)(8,45,25,33)(9,46,26,34)(10,47,27,35)(11,139,159,142)(12,140,160,143)(13,131,151,144)(14,132,152,145)(15,133,153,146)(16,134,154,147)(17,135,155,148)(18,136,156,149)(19,137,157,150)(20,138,158,141)(51,71,63,90)(52,72,64,81)(53,73,65,82)(54,74,66,83)(55,75,67,84)(56,76,68,85)(57,77,69,86)(58,78,70,87)(59,79,61,88)(60,80,62,89)(91,124,104,111)(92,125,105,112)(93,126,106,113)(94,127,107,114)(95,128,108,115)(96,129,109,116)(97,130,110,117)(98,121,101,118)(99,122,102,119)(100,123,103,120), (1,68,28,56)(2,69,29,57)(3,70,30,58)(4,61,21,59)(5,62,22,60)(6,63,23,51)(7,64,24,52)(8,65,25,53)(9,66,26,54)(10,67,27,55)(11,122,159,119)(12,123,160,120)(13,124,151,111)(14,125,152,112)(15,126,153,113)(16,127,154,114)(17,128,155,115)(18,129,156,116)(19,130,157,117)(20,121,158,118)(31,90,43,71)(32,81,44,72)(33,82,45,73)(34,83,46,74)(35,84,47,75)(36,85,48,76)(37,86,49,77)(38,87,50,78)(39,88,41,79)(40,89,42,80)(91,144,104,131)(92,145,105,132)(93,146,106,133)(94,147,107,134)(95,148,108,135)(96,149,109,136)(97,150,110,137)(98,141,101,138)(99,142,102,139)(100,143,103,140), (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,99,6,94)(2,98,7,93)(3,97,8,92)(4,96,9,91)(5,95,10,100)(11,90,16,85)(12,89,17,84)(13,88,18,83)(14,87,19,82)(15,86,20,81)(21,109,26,104)(22,108,27,103)(23,107,28,102)(24,106,29,101)(25,105,30,110)(31,114,36,119)(32,113,37,118)(33,112,38,117)(34,111,39,116)(35,120,40,115)(41,129,46,124)(42,128,47,123)(43,127,48,122)(44,126,49,121)(45,125,50,130)(51,134,56,139)(52,133,57,138)(53,132,58,137)(54,131,59,136)(55,140,60,135)(61,149,66,144)(62,148,67,143)(63,147,68,142)(64,146,69,141)(65,145,70,150)(71,154,76,159)(72,153,77,158)(73,152,78,157)(74,151,79,156)(75,160,80,155) );

G=PermutationGroup([[(1,48,28,36),(2,49,29,37),(3,50,30,38),(4,41,21,39),(5,42,22,40),(6,43,23,31),(7,44,24,32),(8,45,25,33),(9,46,26,34),(10,47,27,35),(11,139,159,142),(12,140,160,143),(13,131,151,144),(14,132,152,145),(15,133,153,146),(16,134,154,147),(17,135,155,148),(18,136,156,149),(19,137,157,150),(20,138,158,141),(51,71,63,90),(52,72,64,81),(53,73,65,82),(54,74,66,83),(55,75,67,84),(56,76,68,85),(57,77,69,86),(58,78,70,87),(59,79,61,88),(60,80,62,89),(91,124,104,111),(92,125,105,112),(93,126,106,113),(94,127,107,114),(95,128,108,115),(96,129,109,116),(97,130,110,117),(98,121,101,118),(99,122,102,119),(100,123,103,120)], [(1,68,28,56),(2,69,29,57),(3,70,30,58),(4,61,21,59),(5,62,22,60),(6,63,23,51),(7,64,24,52),(8,65,25,53),(9,66,26,54),(10,67,27,55),(11,122,159,119),(12,123,160,120),(13,124,151,111),(14,125,152,112),(15,126,153,113),(16,127,154,114),(17,128,155,115),(18,129,156,116),(19,130,157,117),(20,121,158,118),(31,90,43,71),(32,81,44,72),(33,82,45,73),(34,83,46,74),(35,84,47,75),(36,85,48,76),(37,86,49,77),(38,87,50,78),(39,88,41,79),(40,89,42,80),(91,144,104,131),(92,145,105,132),(93,146,106,133),(94,147,107,134),(95,148,108,135),(96,149,109,136),(97,150,110,137),(98,141,101,138),(99,142,102,139),(100,143,103,140)], [(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),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,99,6,94),(2,98,7,93),(3,97,8,92),(4,96,9,91),(5,95,10,100),(11,90,16,85),(12,89,17,84),(13,88,18,83),(14,87,19,82),(15,86,20,81),(21,109,26,104),(22,108,27,103),(23,107,28,102),(24,106,29,101),(25,105,30,110),(31,114,36,119),(32,113,37,118),(33,112,38,117),(34,111,39,116),(35,120,40,115),(41,129,46,124),(42,128,47,123),(43,127,48,122),(44,126,49,121),(45,125,50,130),(51,134,56,139),(52,133,57,138),(53,132,58,137),(54,131,59,136),(55,140,60,135),(61,149,66,144),(62,148,67,143),(63,147,68,142),(64,146,69,141),(65,145,70,150),(71,154,76,159),(72,153,77,158),(73,152,78,157),(74,151,79,156),(75,160,80,155)]])

Q₈×Dic₅ is a maximal subgroup of
Dic₅.₁₂Q₁₆ Dic₅⋊₇SD₁₆ C₅⋊Q₁₆⋊₅C₄ Dic₅⋊₄Q₁₆ Q₈⋊Dic₁₀ Dic₅.₉Q₁₆ Q₈.Dic₁₀ Q₈.₂Dic₁₀ Q₈⋊D₅⋊₆C₄ Dic₅⋊₅SD₁₆ SD₁₆⋊Dic₅ (C₅×Q₈).D₄ Dic₅⋊₃Q₁₆ Q₁₆⋊Dic₅ (C₂×Q₁₆)⋊D₅ C₂₀.₆M₄(2) Q₈⋊₅Dic₁₀ Q₈⋊₆Dic₁₀ C₄×Q₈×D₅ C₄².₁₂₅D₁₀ C₄×Q₈⋊₂D₅ C₄².₁₂₆D₁₀ (Q₈×Dic₅)⋊C₂ C₂²⋊Q₈⋊₂₅D₅ C₁₀.₁₅2_-¹⁺⁴ C₁₀.₁₁₈2₊¹⁺⁴ C₁₀.₂₁2_-¹⁺⁴ C₁₀.₂₃2_-¹⁺⁴ C₁₀.₇₇2_-¹⁺⁴ C₁₀.₂₄2_-¹⁺⁴ C₄².₁₃₉D₁₀ C₄².₂₃₄D₁₀ C₄².₁₄₃D₁₀ C₄².₁₄₄D₁₀ Dic₁₀⋊₈Q₈ C₄².₂₄₁D₁₀ C₄².₁₇₆D₁₀ C₄².₁₇₇D₁₀ C₁₀.₄₂2_-¹⁺⁴ C₁₀.₄₅2_-¹⁺⁴ C₁₀.₁₀₆2_-¹⁺⁴ C₁₀.₁₀₇2_-¹⁺⁴ C₁₀.₁₄₈2₊¹⁺⁴ Dic₁₅⋊₇Q₈
Q₈×Dic₅ is a maximal quotient of
C₄⋊C₄⋊₅Dic₅ C₂₀⋊₆(C₄⋊C₄) C₄².₂₁₀D₁₀ (Q₈×C₁₀)⋊₁₇C₄ Dic₁₅⋊₇Q₈

40 conjugacy classes

class	1	2A	2B	2C	4A	···	4F	4G	4H	4I	4J	4K	···	4P	5A	5B	10A	···	10F	20A	···	20L
order	1	2	2	2	4	···	4	4	4	4	4	4	···	4	5	5	10	···	10	20	···	20
size	1	1	1	1	2	···	2	5	5	5	5	10	···	10	2	2	2	···	2	4	···	4

40 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	4	4
type	+	+	+	+		-	+		+	-	-	+
image	C₁	C₂	C₂	C₂	C₄	Q₈	D₅	C₄○D₄	D₁₀	Dic₅	Q₈×D₅	Q₈⋊₂D₅
kernel	Q₈×Dic₅	C₄×Dic₅	C₄⋊Dic₅	Q₈×C₁₀	C₅×Q₈	Dic₅	C₂×Q₈	C₁₀	C₂×C₄	Q₈	C₂	C₂
# reps	1	3	3	1	8	2	2	2	6	8	2	2

Matrix representation of Q₈×Dic₅ ►in GL₅(𝔽₄₁)

40	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	40	0

40	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	20	38
0	0	0	38	21

40	0	0	0	0
0	0	1	0	0
0	40	6	0	0
0	0	0	1	0
0	0	0	0	1

32	0	0	0	0
0	16	25	0	0
0	39	25	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,0],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,20,38,0,0,0,38,21],[40,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,16,39,0,0,0,25,25,0,0,0,0,0,1,0,0,0,0,0,1] >;

Q₈×Dic₅ in GAP, Magma, Sage, TeX

Q_8\times {\rm Dic}_5

% in TeX

G:=Group("Q8xDic5");

// GroupNames label

G:=SmallGroup(160,166);

// by ID

G=gap.SmallGroup(160,166);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,188,86,4613]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=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 = Q8×Dic5 order 160 = 25·5

Direct product of Q8 and Dic5

G = Q₈×Dic₅ order 160 = 2⁵·5

Direct product of Q₈ and Dic₅