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## G = C20⋊SD16order 320 = 26·5

### 1st semidirect product of C20 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C20⋊SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4×D20 — C20⋊SD16
 Lower central C5 — C10 — C2×C20 — C20⋊SD16
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C20⋊SD16
G = < a,b,c | a20=b8=c2=1, bab-1=a11, cac=a9, cbc=b3 >

Subgroups: 566 in 120 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C20, D10, C2×C10, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4.D4, C40⋊C2, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, C5×C4⋊C8, C202Q8, C4×D20, C2×C40⋊C2, C20⋊SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8.C22, D20, C22×D5, D4.D4, C40⋊C2, C2×D20, D4×D5, Q82D5, C4⋊D20, C2×C40⋊C2, C8.D10, C20⋊SD16

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

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

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

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

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 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 20 20 2 2 2 2 4 20 20 40 40 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D5 SD16 C4○D4 D10 D10 D20 C40⋊C2 C8.C22 D4×D5 Q8⋊2D5 C8.D10 kernel C20⋊SD16 C20.44D4 C5×C4⋊C8 C20⋊2Q8 C4×D20 C2×C40⋊C2 D20 C2×C20 C4⋊C8 C20 C20 C42 C2×C8 C2×C4 C4 C10 C4 C4 C2 # reps 1 2 1 1 1 2 2 2 2 4 2 2 4 8 16 1 2 2 4

Matrix representation of C20⋊SD16 in GL6(𝔽41)

 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 40 24 0 0 0 0 17 1
,
 36 19 0 0 0 0 34 35 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 28 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,40,17,0,0,0,0,24,1],[36,34,0,0,0,0,19,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C20⋊SD16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,468);
// by ID

G=gap.SmallGroup(320,468);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^11,c*a*c=a^9,c*b*c=b^3>;
// generators/relations

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