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## G = (C2×C4)⋊9D20order 320 = 26·5

### 1st semidirect product of C2×C4 and D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×C4)⋊9D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×D20 — (C2×C4)⋊9D20
 Lower central C5 — C2×C10 — (C2×C4)⋊9D20
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4)⋊9D20
G = < a,b,c,d | a2=b4=c20=d2=1, cbc-1=dbd=ab=ba, ac=ca, ad=da, dcd=c-1 >

Subgroups: 1294 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2.C42, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.23D4, D10⋊C4, C2×C4×D5, C2×D20, C2×D20, C22×Dic5, C22×C20, C23×D5, C10.10C42, C5×C2.C42, C2×D10⋊C4, D5×C22×C4, C22×D20, (C2×C4)⋊9D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4×D5, D20, C22×D5, C23.23D4, C2×C4×D5, C2×D20, C4○D20, D4×D5, Q82D5, C4×D20, D5×C22⋊C4, C22⋊D20, D10⋊D4, D208C4, D10.13D4, C4⋊D20, (C2×C4)⋊9D20

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 20 20 2 2 2 2 4 4 4 4 10 10 10 10 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 D5 C4○D4 D10 C4×D5 D20 C4○D20 D4×D5 Q8⋊2D5 kernel (C2×C4)⋊9D20 C10.10C42 C5×C2.C42 C2×D10⋊C4 D5×C22×C4 C22×D20 C2×D20 C2×Dic5 C2×C20 C22×D5 C2.C42 C2×C10 C22×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 1 3 1 1 8 2 2 4 2 4 6 8 8 8 6 2

Matrix representation of (C2×C4)⋊9D20 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
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 9 11 0 0 0 0 30 14 0 0 0 0 0 0 34 40 0 0 0 0 8 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 40 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40

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],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,30,0,0,0,0,11,14,0,0,0,0,0,0,34,8,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

(C2×C4)⋊9D20 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_9D_{20}
% in TeX

G:=Group("(C2xC4):9D20");
// GroupNames label

G:=SmallGroup(320,292);
// by ID

G=gap.SmallGroup(320,292);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,12550]);
// Polycyclic

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

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