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## G = (C2×C4).20D20order 320 = 26·5

### 13rd 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).20D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — (C2×C4).20D20
 Lower central C5 — C22×C10 — (C2×C4).20D20
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4).20D20
G = < a,b,c,d | a2=b20=c4=1, d2=b10, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 886 in 202 conjugacy classes, 61 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], Q8 [×8], C23, C23 [×8], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×Q8 [×6], C24, Dic5 [×5], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×2], C2×C22⋊C4 [×3], C22×Q8, Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C23⋊Q8, D10⋊C4 [×6], C2×Dic10 [×6], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42 [×2], C5×C2.C42, C2×D10⋊C4, C2×D10⋊C4 [×2], C22×Dic10, (C2×C4).20D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], D20 [×2], C22×D5, C23⋊Q8, C2×D20, C4○D20 [×2], D4×D5 [×2], D42D5, Q8×D5, C4.D20, C22⋊D20, Dic5.5D4 [×2], D10⋊Q8 [×2], D102Q8, (C2×C4).20D20

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

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

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

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

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 2 4 4 4 type + + + + + + + - + + + + - - image C1 C2 C2 C2 C2 D4 D4 Q8 D5 C4○D4 D10 D20 C4○D20 D4×D5 D4⋊2D5 Q8×D5 kernel (C2×C4).20D20 C10.10C42 C5×C2.C42 C2×D10⋊C4 C22×Dic10 C2×Dic5 C2×C20 C22×D5 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 2 1 3 1 4 2 2 2 6 6 8 16 4 2 2

Matrix representation of (C2×C4).20D20 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
,
 14 30 0 0 0 0 11 9 0 0 0 0 0 0 34 1 0 0 0 0 33 1 0 0 0 0 0 0 32 0 0 0 0 0 21 9
,
 24 1 0 0 0 0 40 17 0 0 0 0 0 0 11 28 0 0 0 0 22 30 0 0 0 0 0 0 16 2 0 0 0 0 16 25
,
 24 1 0 0 0 0 38 17 0 0 0 0 0 0 14 2 0 0 0 0 5 27 0 0 0 0 0 0 16 2 0 0 0 0 15 25

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],[14,11,0,0,0,0,30,9,0,0,0,0,0,0,34,33,0,0,0,0,1,1,0,0,0,0,0,0,32,21,0,0,0,0,0,9],[24,40,0,0,0,0,1,17,0,0,0,0,0,0,11,22,0,0,0,0,28,30,0,0,0,0,0,0,16,16,0,0,0,0,2,25],[24,38,0,0,0,0,1,17,0,0,0,0,0,0,14,5,0,0,0,0,2,27,0,0,0,0,0,0,16,15,0,0,0,0,2,25] >;

(C2×C4).20D20 in GAP, Magma, Sage, TeX

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

G:=Group("(C2xC4).20D20");
// GroupNames label

G:=SmallGroup(320,300);
// by ID

G=gap.SmallGroup(320,300);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,926,387,268,12550]);
// Polycyclic

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

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