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## G = (C2×Dic5)⋊C8order 320 = 26·5

### 1st semidirect product of C2×Dic5 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×Dic5)⋊C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C22×C20 — C2×C10.D4 — (C2×Dic5)⋊C8
 Lower central C5 — C10 — C2×C10 — (C2×Dic5)⋊C8
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for (C2×Dic5)⋊C8
G = < a,b,c,d | a2=b10=d8=1, c2=b5, ab=ba, ac=ca, dad-1=ab5, cbc-1=b-1, bd=db, dcd-1=ab5c >

Subgroups: 278 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], Dic5 [×3], C20 [×2], C2×C10 [×3], C2×C10 [×2], C22⋊C8, C22⋊C8, C2×C4⋊C4, C52C8, C40, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20, C22×C10, C22.M4(2), C2×C52C8, C10.D4 [×2], C2×C40, C22×Dic5 [×2], C22×C20, C20.55D4, C5×C22⋊C8, C2×C10.D4, (C2×Dic5)⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C23⋊C4, C4.10D4, C4×D5, D20, C5⋊D4, C22.M4(2), C8×D5, C8⋊D5, D10⋊C4, C23.1D10, D101C8, C4.12D20, (C2×Dic5)⋊C8

Smallest permutation representation of (C2×Dic5)⋊C8
On 160 points
Generators in S160
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)(121 126)(122 127)(123 128)(124 129)(125 130)(131 136)(132 137)(133 138)(134 139)(135 140)(141 146)(142 147)(143 148)(144 149)(145 150)(151 156)(152 157)(153 158)(154 159)(155 160)
(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 84 6 89)(2 83 7 88)(3 82 8 87)(4 81 9 86)(5 90 10 85)(11 94 16 99)(12 93 17 98)(13 92 18 97)(14 91 19 96)(15 100 20 95)(21 104 26 109)(22 103 27 108)(23 102 28 107)(24 101 29 106)(25 110 30 105)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 124 46 129)(42 123 47 128)(43 122 48 127)(44 121 49 126)(45 130 50 125)(51 134 56 139)(52 133 57 138)(53 132 58 137)(54 131 59 136)(55 140 60 135)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)
(1 71 31 51 11 61 21 41)(2 72 32 52 12 62 22 42)(3 73 33 53 13 63 23 43)(4 74 34 54 14 64 24 44)(5 75 35 55 15 65 25 45)(6 76 36 56 16 66 26 46)(7 77 37 57 17 67 27 47)(8 78 38 58 18 68 28 48)(9 79 39 59 19 69 29 49)(10 80 40 60 20 70 30 50)(81 156 116 131 91 146 106 121)(82 157 117 132 92 147 107 122)(83 158 118 133 93 148 108 123)(84 159 119 134 94 149 109 124)(85 160 120 135 95 150 110 125)(86 151 111 136 96 141 101 126)(87 152 112 137 97 142 102 127)(88 153 113 138 98 143 103 128)(89 154 114 139 99 144 104 129)(90 155 115 140 100 145 105 130)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 2 2 2 2 20 20 20 20 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - image C1 C2 C2 C2 C4 C8 D4 D5 M4(2) D10 D20 C5⋊D4 C4×D5 C8×D5 C8⋊D5 C23⋊C4 C4.10D4 C23.1D10 C4.12D20 kernel (C2×Dic5)⋊C8 C20.55D4 C5×C22⋊C8 C2×C10.D4 C22×Dic5 C2×Dic5 C2×C20 C22⋊C8 C2×C10 C22×C4 C2×C4 C2×C4 C23 C22 C22 C10 C10 C2 C2 # reps 1 1 1 1 4 8 2 2 2 2 4 4 4 8 8 1 1 4 4

Matrix representation of (C2×Dic5)⋊C8 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 40 0 0 0 0 0 13 1 0 40
,
 3 34 0 0 0 0 0 0 34 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 18 35 0 0 0 0 0 0 6 23 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 27 11 0 0 0 0 0 0 38 14 0 0 0 0 0 0 0 34 11 26 0 0 0 0 26 2 30 30
,
 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 14 0 0 0 0 0 0 14 0 0 0 0 0 0 0 0 0 32 0 37 0 0 0 0 0 0 1 17 39 0 0 0 0 35 1 9 0 0 0 0 0 6 0 3 40

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,16,13,0,0,0,0,0,1,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[3,34,0,0,0,0,0,0,34,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[18,6,0,0,0,0,0,0,35,23,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,27,38,0,26,0,0,0,0,11,14,34,2,0,0,0,0,0,0,11,30,0,0,0,0,0,0,26,30],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,32,0,35,6,0,0,0,0,0,1,1,0,0,0,0,0,37,17,9,3,0,0,0,0,0,39,0,40] >;

(C2×Dic5)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times {\rm Dic}_5)\rtimes C_8
% in TeX

G:=Group("(C2xDic5):C8");
// GroupNames label

G:=SmallGroup(320,27);
// by ID

G=gap.SmallGroup(320,27);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,758,100,570,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=d^8=1,c^2=b^5,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^5,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^5*c>;
// generators/relations

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