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## G = C10×C8○D4order 320 = 26·5

### Direct product of C10 and C8○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C10×C8○D4
 Chief series C1 — C2 — C4 — C20 — C40 — C2×C40 — C5×C8○D4 — C10×C8○D4
 Lower central C1 — C2 — C10×C8○D4
 Upper central C1 — C2×C40 — C10×C8○D4

Generators and relations for C10×C8○D4
G = < a,b,c,d | a10=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 290 in 266 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C2×C8○D4, C2×C40, C2×C40, C5×M4(2), C22×C20, D4×C10, Q8×C10, C5×C4○D4, C22×C40, C10×M4(2), C5×C8○D4, C10×C4○D4, C10×C8○D4
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C20, C2×C10, C8○D4, C23×C4, C2×C20, C22×C10, C2×C8○D4, C22×C20, C23×C10, C5×C8○D4, C23×C20, C10×C8○D4

Smallest permutation representation of C10×C8○D4
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)
(1 125 62 135 54 147 50 119)(2 126 63 136 55 148 41 120)(3 127 64 137 56 149 42 111)(4 128 65 138 57 150 43 112)(5 129 66 139 58 141 44 113)(6 130 67 140 59 142 45 114)(7 121 68 131 60 143 46 115)(8 122 69 132 51 144 47 116)(9 123 70 133 52 145 48 117)(10 124 61 134 53 146 49 118)(11 81 160 91 29 103 39 75)(12 82 151 92 30 104 40 76)(13 83 152 93 21 105 31 77)(14 84 153 94 22 106 32 78)(15 85 154 95 23 107 33 79)(16 86 155 96 24 108 34 80)(17 87 156 97 25 109 35 71)(18 88 157 98 26 110 36 72)(19 89 158 99 27 101 37 73)(20 90 159 100 28 102 38 74)
(1 74 54 100)(2 75 55 91)(3 76 56 92)(4 77 57 93)(5 78 58 94)(6 79 59 95)(7 80 60 96)(8 71 51 97)(9 72 52 98)(10 73 53 99)(11 148 29 126)(12 149 30 127)(13 150 21 128)(14 141 22 129)(15 142 23 130)(16 143 24 121)(17 144 25 122)(18 145 26 123)(19 146 27 124)(20 147 28 125)(31 138 152 112)(32 139 153 113)(33 140 154 114)(34 131 155 115)(35 132 156 116)(36 133 157 117)(37 134 158 118)(38 135 159 119)(39 136 160 120)(40 137 151 111)(41 103 63 81)(42 104 64 82)(43 105 65 83)(44 106 66 84)(45 107 67 85)(46 108 68 86)(47 109 69 87)(48 110 70 88)(49 101 61 89)(50 102 62 90)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 121)(12 122)(13 123)(14 124)(15 125)(16 126)(17 127)(18 128)(19 129)(20 130)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 141)(28 142)(29 143)(30 144)(31 117)(32 118)(33 119)(34 120)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 108)(42 109)(43 110)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 91)(61 84)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 81)(69 82)(70 83)(131 160)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 157)(139 158)(140 159)

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

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

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

200 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 5A 5B 5C 5D 8A ··· 8H 8I ··· 8T 10A ··· 10L 10M ··· 10AJ 20A ··· 20P 20Q ··· 20AN 40A ··· 40AF 40AG ··· 40CB order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 5 5 8 ··· 8 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C20 C20 C20 C8○D4 C5×C8○D4 kernel C10×C8○D4 C22×C40 C10×M4(2) C5×C8○D4 C10×C4○D4 D4×C10 Q8×C10 C5×C4○D4 C2×C8○D4 C22×C8 C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C10 C2 # reps 1 3 3 8 1 6 2 8 4 12 12 32 4 24 8 32 8 32

Matrix representation of C10×C8○D4 in GL3(𝔽41) generated by

 40 0 0 0 16 0 0 0 16
,
 1 0 0 0 14 0 0 0 14
,
 1 0 0 0 0 1 0 40 0
,
 1 0 0 0 0 40 0 40 0
G:=sub<GL(3,GF(41))| [40,0,0,0,16,0,0,0,16],[1,0,0,0,14,0,0,0,14],[1,0,0,0,0,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;

C10×C8○D4 in GAP, Magma, Sage, TeX

C_{10}\times C_8\circ D_4
% in TeX

G:=Group("C10xC8oD4");
// GroupNames label

G:=SmallGroup(320,1569);
// by ID

G=gap.SmallGroup(320,1569);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1731,124]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

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