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## G = C8⋊(C4×D5)  order 320 = 26·5

### 3rd semidirect product of C8 and C4×D5 acting via C4×D5/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C8⋊(C4×D5)
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×C8⋊D5 — C8⋊(C4×D5)
 Lower central C5 — C10 — C20 — C8⋊(C4×D5)
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C8⋊(C4×D5)
G = < a,b,c,d | a8=b4=c5=d2=1, bab-1=a3, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 430 in 118 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.Q8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C52C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, M4(2)⋊C4, C8⋊D5, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C10.D8, C20.Q8, C405C4, C5×C4.Q8, D5×C4⋊C4, C4⋊C47D5, C2×C8⋊D5, C8⋊(C4×D5)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C8⋊C22, C8.C22, C4×D5, C22×D5, M4(2)⋊C4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D40⋊C2, SD16⋊D5, C8⋊(C4×D5)

Smallest permutation representation of C8⋊(C4×D5)
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 55 25 155)(2 50 26 158)(3 53 27 153)(4 56 28 156)(5 51 29 159)(6 54 30 154)(7 49 31 157)(8 52 32 160)(9 143 63 21)(10 138 64 24)(11 141 57 19)(12 144 58 22)(13 139 59 17)(14 142 60 20)(15 137 61 23)(16 140 62 18)(33 89 66 126)(34 92 67 121)(35 95 68 124)(36 90 69 127)(37 93 70 122)(38 96 71 125)(39 91 72 128)(40 94 65 123)(41 108 76 130)(42 111 77 133)(43 106 78 136)(44 109 79 131)(45 112 80 134)(46 107 73 129)(47 110 74 132)(48 105 75 135)(81 152 113 99)(82 147 114 102)(83 150 115 97)(84 145 116 100)(85 148 117 103)(86 151 118 98)(87 146 119 101)(88 149 120 104)
(1 103 133 21 39)(2 104 134 22 40)(3 97 135 23 33)(4 98 136 24 34)(5 99 129 17 35)(6 100 130 18 36)(7 101 131 19 37)(8 102 132 20 38)(9 91 55 85 42)(10 92 56 86 43)(11 93 49 87 44)(12 94 50 88 45)(13 95 51 81 46)(14 96 52 82 47)(15 89 53 83 48)(16 90 54 84 41)(25 148 111 143 72)(26 149 112 144 65)(27 150 105 137 66)(28 151 106 138 67)(29 152 107 139 68)(30 145 108 140 69)(31 146 109 141 70)(32 147 110 142 71)(57 122 157 119 79)(58 123 158 120 80)(59 124 159 113 73)(60 125 160 114 74)(61 126 153 115 75)(62 127 154 116 76)(63 128 155 117 77)(64 121 156 118 78)
(1 39)(2 36)(3 33)(4 38)(5 35)(6 40)(7 37)(8 34)(9 85)(10 82)(11 87)(12 84)(13 81)(14 86)(15 83)(16 88)(17 99)(18 104)(19 101)(20 98)(21 103)(22 100)(23 97)(24 102)(25 72)(26 69)(27 66)(28 71)(29 68)(30 65)(31 70)(32 67)(41 45)(43 47)(49 93)(50 90)(51 95)(52 92)(53 89)(54 94)(55 91)(56 96)(57 119)(58 116)(59 113)(60 118)(61 115)(62 120)(63 117)(64 114)(74 78)(76 80)(106 110)(108 112)(121 160)(122 157)(123 154)(124 159)(125 156)(126 153)(127 158)(128 155)(130 134)(132 136)(137 150)(138 147)(139 152)(140 149)(141 146)(142 151)(143 148)(144 145)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - + + + + + + - - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 Q8 D4 D4 D5 D10 D10 C4×D5 C8⋊C22 C8.C22 Q8×D5 D4×D5 D40⋊C2 SD16⋊D5 kernel C8⋊(C4×D5) C10.D8 C20.Q8 C40⋊5C4 C5×C4.Q8 D5×C4⋊C4 C4⋊C4⋊7D5 C2×C8⋊D5 C8⋊D5 C4×D5 C2×Dic5 C22×D5 C4.Q8 C4⋊C4 C2×C8 C8 C10 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 2 4 2 8 1 1 2 2 4 4

Matrix representation of C8⋊(C4×D5) in GL6(𝔽41)

 35 2 0 0 0 0 2 6 0 0 0 0 0 0 32 15 9 26 0 0 26 9 15 32 0 0 32 15 32 15 0 0 26 9 26 9
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 27 0 7 0 0 0 0 27 0 7 0 0 7 0 14 0 0 0 0 7 0 14
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 40 0 0 0 0 1 0 0 0 0 0 0 0 34 40 0 0 0 0 1 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 34 40 0 0 0 0 7 7 0 0 0 0 0 0 34 40 0 0 0 0 7 7

G:=sub<GL(6,GF(41))| [35,2,0,0,0,0,2,6,0,0,0,0,0,0,32,26,32,26,0,0,15,9,15,9,0,0,9,15,32,26,0,0,26,32,15,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,27,0,7,0,0,0,0,27,0,7,0,0,7,0,14,0,0,0,0,7,0,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,34,7,0,0,0,0,40,7] >;

C8⋊(C4×D5) in GAP, Magma, Sage, TeX

C_8\rtimes (C_4\times D_5)
% in TeX

G:=Group("C8:(C4xD5)");
// GroupNames label

G:=SmallGroup(320,488);
// by ID

G=gap.SmallGroup(320,488);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,555,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^4=c^5=d^2=1,b*a*b^-1=a^3,a*c=c*a,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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