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## G = (C2×C4).21D20order 320 = 26·5

### 14th non-split extension by C2×C4 of D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×C4).21D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — (C2×C4).21D20
 Lower central C5 — C22×C10 — (C2×C4).21D20
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4).21D20
G = < a,b,c,d | a2=b4=c20=1, d2=ab2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=ab2c-1 >

Subgroups: 742 in 170 conjugacy classes, 57 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×17], C23, C23 [×8], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×3], C24, Dic5 [×3], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×2], C2×C22⋊C4 [×3], C2×C4⋊C4, C2×Dic5 [×9], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.11D4, C4⋊Dic5 [×2], D10⋊C4 [×6], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42 [×2], C5×C2.C42, C2×C4⋊Dic5, C2×D10⋊C4, C2×D10⋊C4 [×2], (C2×C4).21D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4 [×5], D10 [×3], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], D20 [×2], C22×D5, C23.11D4, C2×D20, C4○D20 [×2], D4×D5, D42D5 [×2], Q82D5, C4.D20, D10.12D4 [×2], C22.D20, C4⋊D20, C4⋊C4⋊D5 [×2], (C2×C4).21D20

Smallest permutation representation of (C2×C4).21D20
On 160 points
Generators in S160
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 141)(18 142)(19 143)(20 144)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 116)(34 117)(35 118)(36 119)(37 120)(38 101)(39 102)(40 103)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 61)(81 140)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 128)(90 129)(91 130)(92 131)(93 132)(94 133)(95 134)(96 135)(97 136)(98 137)(99 138)(100 139)
(1 61 95 28)(2 41 96 112)(3 63 97 30)(4 43 98 114)(5 65 99 32)(6 45 100 116)(7 67 81 34)(8 47 82 118)(9 69 83 36)(10 49 84 120)(11 71 85 38)(12 51 86 102)(13 73 87 40)(14 53 88 104)(15 75 89 22)(16 55 90 106)(17 77 91 24)(18 57 92 108)(19 79 93 26)(20 59 94 110)(21 158 74 127)(23 160 76 129)(25 142 78 131)(27 144 80 133)(29 146 62 135)(31 148 64 137)(33 150 66 139)(35 152 68 121)(37 154 70 123)(39 156 72 125)(42 136 113 147)(44 138 115 149)(46 140 117 151)(48 122 119 153)(50 124 101 155)(52 126 103 157)(54 128 105 159)(56 130 107 141)(58 132 109 143)(60 134 111 145)
(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)
(1 133 134 20)(2 19 135 132)(3 131 136 18)(4 17 137 130)(5 129 138 16)(6 15 139 128)(7 127 140 14)(8 13 121 126)(9 125 122 12)(10 11 123 124)(21 46 53 34)(22 33 54 45)(23 44 55 32)(24 31 56 43)(25 42 57 30)(26 29 58 41)(27 60 59 28)(35 52 47 40)(36 39 48 51)(37 50 49 38)(61 80 111 110)(62 109 112 79)(63 78 113 108)(64 107 114 77)(65 76 115 106)(66 105 116 75)(67 74 117 104)(68 103 118 73)(69 72 119 102)(70 101 120 71)(81 158 151 88)(82 87 152 157)(83 156 153 86)(84 85 154 155)(89 150 159 100)(90 99 160 149)(91 148 141 98)(92 97 142 147)(93 146 143 96)(94 95 144 145)

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4F 4G ··· 4L 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 4 ··· 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + image C1 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D20 C4○D20 D4×D5 D4⋊2D5 Q8⋊2D5 kernel (C2×C4).21D20 C10.10C42 C5×C2.C42 C2×C4⋊Dic5 C2×D10⋊C4 C2×C20 C22×D5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 2 1 1 3 2 2 2 10 6 8 16 2 4 2

Matrix representation of (C2×C4).21D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 32 0 0 0 0 9 11 0 0 0 0 0 0 9 0 0 0 0 0 0 32
,
 32 30 0 0 0 0 11 27 0 0 0 0 0 0 0 32 0 0 0 0 9 22 0 0 0 0 0 0 0 32 0 0 0 0 32 0
,
 9 11 0 0 0 0 30 32 0 0 0 0 0 0 0 32 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 9 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[32,11,0,0,0,0,30,27,0,0,0,0,0,0,0,9,0,0,0,0,32,22,0,0,0,0,0,0,0,32,0,0,0,0,32,0],[9,30,0,0,0,0,11,32,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,32,0] >;

(C2×C4).21D20 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{21}D_{20}
% in TeX

G:=Group("(C2xC4).21D20");
// GroupNames label

G:=SmallGroup(320,301);
// by ID

G=gap.SmallGroup(320,301);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,254,387,100,12550]);
// Polycyclic

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

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