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## G = (C2×C4)⋊3D20order 320 = 26·5

### 2nd semidirect product of C2×C4 and D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×C4)⋊3D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×D20 — (C2×C4)⋊3D20
 Lower central C5 — C22×C10 — (C2×C4)⋊3D20
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C4)⋊3D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=c-1 >

Subgroups: 1158 in 238 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×20], C5, C2×C4 [×4], C2×C4 [×13], D4 [×8], C23, C23 [×16], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×6], C24 [×2], Dic5 [×2], C20 [×5], D10 [×20], C2×C10 [×3], C2×C10 [×4], C2.C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, D20 [×8], C2×Dic5 [×6], C2×C20 [×4], C2×C20 [×7], C22×D5 [×4], C22×D5 [×12], C22×C10, C23.10D4, D10⋊C4 [×8], C5×C4⋊C4 [×2], C2×D20 [×6], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], C10.10C42, C2×D10⋊C4 [×4], C10×C4⋊C4, C22×D20, (C2×C4)⋊3D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, D5, C2×D4 [×4], C4○D4 [×3], D10 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, D20 [×2], C5⋊D4 [×2], C22×D5, C23.10D4, C2×D20, C4○D20, D4×D5 [×2], Q82D5 [×2], C2×C5⋊D4, D10.13D4 [×2], C4⋊D20 [×2], C207D4, C23⋊D10, C20.23D4, (C2×C4)⋊3D20

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

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4F 4G 4H 4I 4J 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 20 20 20 20 4 ··· 4 20 20 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 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D20 C5⋊D4 C4○D20 D4×D5 Q8⋊2D5 kernel (C2×C4)⋊3D20 C10.10C42 C2×D10⋊C4 C10×C4⋊C4 C22×D20 C2×C20 C22×D5 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 4 1 1 4 4 2 6 6 8 8 8 4 4

Matrix representation of (C2×C4)⋊3D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 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 1
,
 16 9 0 0 0 0 40 25 0 0 0 0 0 0 12 37 0 0 0 0 26 29 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 21 40 0 0 0 0 32 20 0 0 0 0 0 0 15 36 0 0 0 0 37 26 0 0 0 0 0 0 28 39 0 0 0 0 2 16
,
 40 0 0 0 0 0 40 1 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 0 0 0 0 0 40 0 0 0 0 40 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,40,0,0,0,0,9,25,0,0,0,0,0,0,12,26,0,0,0,0,37,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[21,32,0,0,0,0,40,20,0,0,0,0,0,0,15,37,0,0,0,0,36,26,0,0,0,0,0,0,28,2,0,0,0,0,39,16],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

(C2×C4)⋊3D20 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3D_{20}
% in TeX

G:=Group("(C2xC4):3D20");
// GroupNames label

G:=SmallGroup(320,618);
// by ID

G=gap.SmallGroup(320,618);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,184,12550]);
// Polycyclic

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

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