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## G = Dic5⋊3Q8order 160 = 25·5

### The semidirect product of Dic5 and Q8 acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5⋊3Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊3Q8
 Lower central C5 — C10 — Dic5⋊3Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for Dic53Q8
G = < a,b,c,d | a10=c4=1, b2=a5, d2=c2, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 168 in 70 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic5, C4×Dic5, C10.D4, C5×C4⋊C4, C2×Dic10, Dic53Q8
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, C22×C4, C2×Q8, C4○D4, D10, C4×Q8, C4×D5, C22×D5, C2×C4×D5, D42D5, Q8×D5, Dic53Q8

Smallest permutation representation of Dic53Q8
Regular action on 160 points
Generators in S160
(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 96 6 91)(2 95 7 100)(3 94 8 99)(4 93 9 98)(5 92 10 97)(11 84 16 89)(12 83 17 88)(13 82 18 87)(14 81 19 86)(15 90 20 85)(21 114 26 119)(22 113 27 118)(23 112 28 117)(24 111 29 116)(25 120 30 115)(31 104 36 109)(32 103 37 108)(33 102 38 107)(34 101 39 106)(35 110 40 105)(41 134 46 139)(42 133 47 138)(43 132 48 137)(44 131 49 136)(45 140 50 135)(51 124 56 129)(52 123 57 128)(53 122 58 127)(54 121 59 126)(55 130 60 125)(61 154 66 159)(62 153 67 158)(63 152 68 157)(64 151 69 156)(65 160 70 155)(71 144 76 149)(72 143 77 148)(73 142 78 147)(74 141 79 146)(75 150 80 145)
(1 101 19 111)(2 110 20 120)(3 109 11 119)(4 108 12 118)(5 107 13 117)(6 106 14 116)(7 105 15 115)(8 104 16 114)(9 103 17 113)(10 102 18 112)(21 94 31 84)(22 93 32 83)(23 92 33 82)(24 91 34 81)(25 100 35 90)(26 99 36 89)(27 98 37 88)(28 97 38 87)(29 96 39 86)(30 95 40 85)(41 159 51 149)(42 158 52 148)(43 157 53 147)(44 156 54 146)(45 155 55 145)(46 154 56 144)(47 153 57 143)(48 152 58 142)(49 151 59 141)(50 160 60 150)(61 124 71 134)(62 123 72 133)(63 122 73 132)(64 121 74 131)(65 130 75 140)(66 129 76 139)(67 128 77 138)(68 127 78 137)(69 126 79 136)(70 125 80 135)
(1 59 19 49)(2 60 20 50)(3 51 11 41)(4 52 12 42)(5 53 13 43)(6 54 14 44)(7 55 15 45)(8 56 16 46)(9 57 17 47)(10 58 18 48)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)

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

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

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

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 1 2 2 2 2 2 4 4 type + + + + + - + + - - image C1 C2 C2 C2 C2 C4 Q8 D5 C4○D4 D10 C4×D5 D4⋊2D5 Q8×D5 kernel Dic5⋊3Q8 C4×Dic5 C10.D4 C5×C4⋊C4 C2×Dic10 Dic10 Dic5 C4⋊C4 C10 C2×C4 C4 C2 C2 # reps 1 3 2 1 1 8 2 2 2 6 8 2 2

Matrix representation of Dic53Q8 in GL4(𝔽41) generated by

 0 40 0 0 1 35 0 0 0 0 1 0 0 0 0 1
,
 28 9 0 0 13 13 0 0 0 0 1 0 0 0 0 1
,
 6 40 0 0 35 35 0 0 0 0 1 21 0 0 37 40
,
 1 0 0 0 0 1 0 0 0 0 8 1 0 0 17 33
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,1,0,0,0,0,1],[28,13,0,0,9,13,0,0,0,0,1,0,0,0,0,1],[6,35,0,0,40,35,0,0,0,0,1,37,0,0,21,40],[1,0,0,0,0,1,0,0,0,0,8,17,0,0,1,33] >;

Dic53Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_3Q_8
% in TeX

G:=Group("Dic5:3Q8");
// GroupNames label

G:=SmallGroup(160,108);
// by ID

G=gap.SmallGroup(160,108);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,116,122,4613]);
// Polycyclic

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

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