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## G = SD16×C20order 320 = 26·5

### Direct product of C20 and SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C20
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×Q8⋊C4 — SD16×C20
 Lower central C1 — C2 — C4 — SD16×C20
 Upper central C1 — C2×C20 — C4×C20 — SD16×C20

Generators and relations for SD16×C20
G = < a,b,c | a20=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 202 in 122 conjugacy classes, 74 normal (50 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C20, C20, C20, C2×C10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C4×SD16, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C22×C20, D4×C10, Q8×C10, C4×C40, C5×D4⋊C4, C5×Q8⋊C4, C5×C4.Q8, D4×C20, Q8×C20, C10×SD16, SD16×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, SD16, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C2×SD16, C4○D8, C2×C20, C5×D4, C22×C10, C4×SD16, C5×SD16, C22×C20, D4×C10, C5×C4○D4, D4×C20, C10×SD16, C5×C4○D8, SD16×C20

Smallest permutation representation of SD16×C20
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)
(1 107 90 124 49 34 80 146)(2 108 91 125 50 35 61 147)(3 109 92 126 51 36 62 148)(4 110 93 127 52 37 63 149)(5 111 94 128 53 38 64 150)(6 112 95 129 54 39 65 151)(7 113 96 130 55 40 66 152)(8 114 97 131 56 21 67 153)(9 115 98 132 57 22 68 154)(10 116 99 133 58 23 69 155)(11 117 100 134 59 24 70 156)(12 118 81 135 60 25 71 157)(13 119 82 136 41 26 72 158)(14 120 83 137 42 27 73 159)(15 101 84 138 43 28 74 160)(16 102 85 139 44 29 75 141)(17 103 86 140 45 30 76 142)(18 104 87 121 46 31 77 143)(19 105 88 122 47 32 78 144)(20 106 89 123 48 33 79 145)
(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(101 138)(102 139)(103 140)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 129)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20AF 20AG ··· 20BD 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 ··· 4 1 1 1 1 2 ··· 2 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C10 C10 C20 D4 SD16 C4○D4 C4○D8 C5×D4 C5×SD16 C5×C4○D4 C5×C4○D8 kernel SD16×C20 C4×C40 C5×D4⋊C4 C5×Q8⋊C4 C5×C4.Q8 D4×C20 Q8×C20 C10×SD16 C5×SD16 C4×SD16 C4×C8 D4⋊C4 Q8⋊C4 C4.Q8 C4×D4 C4×Q8 C2×SD16 SD16 C2×C20 C20 C20 C10 C2×C4 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 4 4 4 4 4 4 4 4 32 2 4 2 4 8 16 8 16

Matrix representation of SD16×C20 in GL3(𝔽41) generated by

 32 0 0 0 18 0 0 0 18
,
 40 0 0 0 15 15 0 26 15
,
 40 0 0 0 1 0 0 0 40
G:=sub<GL(3,GF(41))| [32,0,0,0,18,0,0,0,18],[40,0,0,0,15,26,0,15,15],[40,0,0,0,1,0,0,0,40] >;

SD16×C20 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{20}
% in TeX

G:=Group("SD16xC20");
// GroupNames label

G:=SmallGroup(320,939);
// by ID

G=gap.SmallGroup(320,939);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,436,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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