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## G = C20.(C2×D4)  order 320 = 26·5

### 125th non-split extension by C20 of C2×D4 acting via C2×D4/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C20.(C2×D4)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C23.21D10 — C20.(C2×D4)
 Lower central C5 — C10 — C20 — C20.(C2×D4)
 Upper central C1 — C2×C4 — C22×C4 — C2×C4○D4

Generators and relations for C20.(C2×D4)
G = < a,b,c,d | a20=c4=d2=1, b2=a10, ab=ba, cac-1=a-1, dad=a11, bc=cb, bd=db, dcd=a5c-1 >

Subgroups: 398 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, C20, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C52C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C23.24D4, C2×C52C8, C2×C52C8, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, D4⋊Dic5, Q8⋊Dic5, C22×C52C8, C23.21D10, C10×C4○D4, C20.(C2×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C4○D8, C2×Dic5, C5⋊D4, C22×D5, C23.24D4, C23.D5, C22×Dic5, C2×C5⋊D4, D4.8D10, C2×C23.D5, C20.(C2×D4)

Smallest permutation representation of C20.(C2×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 42 11 52)(2 43 12 53)(3 44 13 54)(4 45 14 55)(5 46 15 56)(6 47 16 57)(7 48 17 58)(8 49 18 59)(9 50 19 60)(10 51 20 41)(21 134 31 124)(22 135 32 125)(23 136 33 126)(24 137 34 127)(25 138 35 128)(26 139 36 129)(27 140 37 130)(28 121 38 131)(29 122 39 132)(30 123 40 133)(61 107 71 117)(62 108 72 118)(63 109 73 119)(64 110 74 120)(65 111 75 101)(66 112 76 102)(67 113 77 103)(68 114 78 104)(69 115 79 105)(70 116 80 106)(81 146 91 156)(82 147 92 157)(83 148 93 158)(84 149 94 159)(85 150 95 160)(86 151 96 141)(87 152 97 142)(88 153 98 143)(89 154 99 144)(90 155 100 145)
(1 143 29 102)(2 142 30 101)(3 141 31 120)(4 160 32 119)(5 159 33 118)(6 158 34 117)(7 157 35 116)(8 156 36 115)(9 155 37 114)(10 154 38 113)(11 153 39 112)(12 152 40 111)(13 151 21 110)(14 150 22 109)(15 149 23 108)(16 148 24 107)(17 147 25 106)(18 146 26 105)(19 145 27 104)(20 144 28 103)(41 89 121 67)(42 88 122 66)(43 87 123 65)(44 86 124 64)(45 85 125 63)(46 84 126 62)(47 83 127 61)(48 82 128 80)(49 81 129 79)(50 100 130 78)(51 99 131 77)(52 98 132 76)(53 97 133 75)(54 96 134 74)(55 95 135 73)(56 94 136 72)(57 93 137 71)(58 92 138 70)(59 91 139 69)(60 90 140 68)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 26)(22 37)(23 28)(24 39)(25 30)(27 32)(29 34)(31 36)(33 38)(35 40)(41 56)(42 47)(43 58)(44 49)(45 60)(46 51)(48 53)(50 55)(52 57)(54 59)(61 83)(62 94)(63 85)(64 96)(65 87)(66 98)(67 89)(68 100)(69 91)(70 82)(71 93)(72 84)(73 95)(74 86)(75 97)(76 88)(77 99)(78 90)(79 81)(80 92)(101 142)(102 153)(103 144)(104 155)(105 146)(106 157)(107 148)(108 159)(109 150)(110 141)(111 152)(112 143)(113 154)(114 145)(115 156)(116 147)(117 158)(118 149)(119 160)(120 151)(121 136)(122 127)(123 138)(124 129)(125 140)(126 131)(128 133)(130 135)(132 137)(134 139)

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

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

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

68 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 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T 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 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 20 20 20 20 2 2 10 ··· 10 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D5 D10 D10 D10 Dic5 C4○D8 C5⋊D4 C5⋊D4 D4.8D10 kernel C20.(C2×D4) D4⋊Dic5 Q8⋊Dic5 C22×C5⋊2C8 C23.21D10 C10×C4○D4 C5×C4○D4 C2×C20 C22×C10 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C10 C2×C4 C23 C2 # reps 1 2 2 1 1 1 8 3 1 2 2 2 2 8 8 12 4 8

Matrix representation of C20.(C2×D4) in GL4(𝔽41) generated by

 23 0 0 0 0 25 0 0 0 0 0 40 0 0 1 0
,
 40 0 0 0 0 40 0 0 0 0 32 0 0 0 0 32
,
 0 1 0 0 40 0 0 0 0 0 12 29 0 0 29 29
,
 40 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [23,0,0,0,0,25,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[0,40,0,0,1,0,0,0,0,0,12,29,0,0,29,29],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;

C20.(C2×D4) in GAP, Magma, Sage, TeX

C_{20}.(C_2\times D_4)
% in TeX

G:=Group("C20.(C2xD4)");
// GroupNames label

G:=SmallGroup(320,860);
// by ID

G=gap.SmallGroup(320,860);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,254,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^11,b*c=c*b,b*d=d*b,d*c*d=a^5*c^-1>;
// generators/relations

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