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## G = C20.38SD16order 320 = 26·5

### 4th non-split extension by C20 of SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20.38SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C20⋊2Q8 — C20.38SD16
 Lower central C5 — C10 — C2×C20 — C20.38SD16
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for C20.38SD16
G = < a,b,c | a20=b8=c2=1, bab-1=a-1, ac=ca, cbc=a10b3 >

Subgroups: 358 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C52C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D42Q8, C2×C52C8, C4⋊Dic5, C4⋊Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, C203C8, C20.Q8, D4⋊Dic5, C202Q8, D4×C20, C20.38SD16
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8⋊C22, Dic10, C5⋊D4, C22×D5, D42Q8, D4.D5, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, C2×D4.D5, D4⋊D10, C20.38SD16

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

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + - + - + image C1 C2 C2 C2 C2 C2 D4 Q8 D5 SD16 C4○D4 D10 D10 D10 C5⋊D4 Dic10 C4○D20 C8⋊C22 D4.D5 D4⋊D10 kernel C20.38SD16 C20⋊3C8 C20.Q8 D4⋊Dic5 C20⋊2Q8 D4×C20 C2×C20 C5×D4 C4×D4 C20 C20 C42 C4⋊C4 C2×D4 C2×C4 D4 C4 C10 C4 C2 # reps 1 1 2 2 1 1 2 2 2 4 2 2 2 2 8 8 8 1 4 4

Matrix representation of C20.38SD16 in GL4(𝔽41) generated by

 20 0 0 0 4 39 0 0 0 0 1 0 0 0 0 1
,
 11 1 0 0 3 30 0 0 0 0 19 32 0 0 4 11
,
 1 0 0 0 19 40 0 0 0 0 1 0 0 0 10 40
G:=sub<GL(4,GF(41))| [20,4,0,0,0,39,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,1,30,0,0,0,0,19,4,0,0,32,11],[1,19,0,0,0,40,0,0,0,0,1,10,0,0,0,40] >;

C20.38SD16 in GAP, Magma, Sage, TeX

C_{20}._{38}{\rm SD}_{16}
% in TeX

G:=Group("C20.38SD16");
// GroupNames label

G:=SmallGroup(320,635);
// by ID

G=gap.SmallGroup(320,635);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,336,253,120,254,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=a^10*b^3>;
// generators/relations

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