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## G = C8×D20order 320 = 26·5

### Direct product of C8 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C8×D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C4×D20 — C8×D20
 Lower central C5 — C10 — C8×D20
 Upper central C1 — C2×C8 — C4×C8

Generators and relations for C8×D20
G = < a,b,c | a8=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 446 in 134 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, C20, D10, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C52C8, C40, C40, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C8×D4, C8×D5, C2×C52C8, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C203C8, D101C8, C4×C40, C4×D20, D5×C2×C8, C8×D20
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D5, C2×C8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C22×C8, C8○D4, C4×D5, D20, C22×D5, C8×D4, C8×D5, C2×C4×D5, C2×D20, C4○D20, C4×D20, D5×C2×C8, D20.3C4, C8×D20

Smallest permutation representation of C8×D20
On 160 points
Generators in S160
(1 39 125 110 142 42 66 98)(2 40 126 111 143 43 67 99)(3 21 127 112 144 44 68 100)(4 22 128 113 145 45 69 81)(5 23 129 114 146 46 70 82)(6 24 130 115 147 47 71 83)(7 25 131 116 148 48 72 84)(8 26 132 117 149 49 73 85)(9 27 133 118 150 50 74 86)(10 28 134 119 151 51 75 87)(11 29 135 120 152 52 76 88)(12 30 136 101 153 53 77 89)(13 31 137 102 154 54 78 90)(14 32 138 103 155 55 79 91)(15 33 139 104 156 56 80 92)(16 34 140 105 157 57 61 93)(17 35 121 106 158 58 62 94)(18 36 122 107 159 59 63 95)(19 37 123 108 160 60 64 96)(20 38 124 109 141 41 65 97)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 31)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 89)(82 88)(83 87)(84 86)(90 100)(91 99)(92 98)(93 97)(94 96)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)(121 123)(124 140)(125 139)(126 138)(127 137)(128 136)(129 135)(130 134)(131 133)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 160)

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

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

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

104 conjugacy classes

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

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 D4 D5 C4○D4 D10 D10 C8○D4 D20 C4×D5 C8×D5 C4○D20 D20.3C4 kernel C8×D20 C20⋊3C8 D10⋊1C8 C4×C40 C4×D20 D5×C2×C8 C4⋊Dic5 D10⋊C4 C2×D20 D20 C40 C4×C8 C20 C42 C2×C8 C10 C8 C2×C4 C4 C4 C2 # reps 1 1 2 1 1 2 2 4 2 16 2 2 2 2 4 4 8 8 16 8 16

Matrix representation of C8×D20 in GL3(𝔽41) generated by

 3 0 0 0 3 0 0 0 3
,
 40 0 0 0 28 39 0 2 16
,
 40 0 0 0 0 40 0 40 0
G:=sub<GL(3,GF(41))| [3,0,0,0,3,0,0,0,3],[40,0,0,0,28,2,0,39,16],[40,0,0,0,0,40,0,40,0] >;

C8×D20 in GAP, Magma, Sage, TeX

C_8\times D_{20}
% in TeX

G:=Group("C8xD20");
// GroupNames label

G:=SmallGroup(320,313);
// by ID

G=gap.SmallGroup(320,313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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