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## G = Dic5.Q8order 160 = 25·5

### 1st non-split extension by Dic5 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic5.Q8
G = < a,b,c,d | a10=c4=1, b2=a5, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a5b, bd=db, dcd-1=a5c-1 >

Subgroups: 144 in 56 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C10, C42, C4⋊C4, C4⋊C4, Dic5, Dic5, C20, C2×C10, C42.C2, C2×Dic5, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, Dic5.Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, D10, C42.C2, C22×D5, C4○D20, D42D5, Q8×D5, Dic5.Q8

Smallest permutation representation of Dic5.Q8
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)
(1 131 6 136)(2 140 7 135)(3 139 8 134)(4 138 9 133)(5 137 10 132)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)(21 143 26 148)(22 142 27 147)(23 141 28 146)(24 150 29 145)(25 149 30 144)(41 153 46 158)(42 152 47 157)(43 151 48 156)(44 160 49 155)(45 159 50 154)(51 99 56 94)(52 98 57 93)(53 97 58 92)(54 96 59 91)(55 95 60 100)(61 104 66 109)(62 103 67 108)(63 102 68 107)(64 101 69 106)(65 110 70 105)(71 119 76 114)(72 118 77 113)(73 117 78 112)(74 116 79 111)(75 115 80 120)(81 121 86 126)(82 130 87 125)(83 129 88 124)(84 128 89 123)(85 127 90 122)
(1 34 27 43)(2 35 28 44)(3 36 29 45)(4 37 30 46)(5 38 21 47)(6 39 22 48)(7 40 23 49)(8 31 24 50)(9 32 25 41)(10 33 26 42)(11 147 156 131)(12 148 157 132)(13 149 158 133)(14 150 159 134)(15 141 160 135)(16 142 151 136)(17 143 152 137)(18 144 153 138)(19 145 154 139)(20 146 155 140)(51 71 70 87)(52 72 61 88)(53 73 62 89)(54 74 63 90)(55 75 64 81)(56 76 65 82)(57 77 66 83)(58 78 67 84)(59 79 68 85)(60 80 69 86)(91 116 107 122)(92 117 108 123)(93 118 109 124)(94 119 110 125)(95 120 101 126)(96 111 102 127)(97 112 103 128)(98 113 104 129)(99 114 105 130)(100 115 106 121)
(1 96 27 102)(2 95 28 101)(3 94 29 110)(4 93 30 109)(5 92 21 108)(6 91 22 107)(7 100 23 106)(8 99 24 105)(9 98 25 104)(10 97 26 103)(11 90 156 74)(12 89 157 73)(13 88 158 72)(14 87 159 71)(15 86 160 80)(16 85 151 79)(17 84 152 78)(18 83 153 77)(19 82 154 76)(20 81 155 75)(31 125 50 119)(32 124 41 118)(33 123 42 117)(34 122 43 116)(35 121 44 115)(36 130 45 114)(37 129 46 113)(38 128 47 112)(39 127 48 111)(40 126 49 120)(51 145 70 139)(52 144 61 138)(53 143 62 137)(54 142 63 136)(55 141 64 135)(56 150 65 134)(57 149 66 133)(58 148 67 132)(59 147 68 131)(60 146 69 140)

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

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

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

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + - + + - - image C1 C2 C2 C2 C2 Q8 D5 C4○D4 D10 C4○D20 D4⋊2D5 Q8×D5 kernel Dic5.Q8 C4×Dic5 C10.D4 C4⋊Dic5 C5×C4⋊C4 Dic5 C4⋊C4 C10 C2×C4 C2 C2 C2 # reps 1 1 4 1 1 2 2 4 6 8 2 2

Matrix representation of Dic5.Q8 in GL4(𝔽41) generated by

 4 0 0 0 39 31 0 0 0 0 1 0 0 0 0 1
,
 18 3 0 0 1 23 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 12 1 0 0 0 0 0 1 0 0 40 0
,
 2 14 0 0 32 39 0 0 0 0 7 27 0 0 27 34
G:=sub<GL(4,GF(41))| [4,39,0,0,0,31,0,0,0,0,1,0,0,0,0,1],[18,1,0,0,3,23,0,0,0,0,40,0,0,0,0,40],[40,12,0,0,0,1,0,0,0,0,0,40,0,0,1,0],[2,32,0,0,14,39,0,0,0,0,7,27,0,0,27,34] >;

Dic5.Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_5.Q_8
% in TeX

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

G:=SmallGroup(160,110);
// by ID

G=gap.SmallGroup(160,110);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,218,188,86,4613]);
// Polycyclic

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

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