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## G = C2×C20⋊7D4order 320 = 26·5

### Direct product of C2 and C20⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×C20⋊7D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C2×C20⋊7D4
 Lower central C5 — C2×C10 — C2×C20⋊7D4
 Upper central C1 — C23 — C23×C4

Generators and relations for C2×C207D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1646 in 426 conjugacy classes, 143 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C4⋊D4, C4⋊Dic5, D10⋊C4, C2×D20, C2×D20, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C22×C20, C22×C20, C23×D5, C23×C10, C2×C4⋊Dic5, C2×D10⋊C4, C207D4, C22×D20, C22×C5⋊D4, C23×C20, C2×C207D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4⋊D4, C22×D4, C2×C4○D4, D20, C5⋊D4, C22×D5, C2×C4⋊D4, C2×D20, C4○D20, C2×C5⋊D4, C23×D5, C207D4, C22×D20, C2×C4○D20, C22×C5⋊D4, C2×C207D4

Smallest permutation representation of C2×C207D4
On 160 points
Generators in S160
(1 155)(2 156)(3 157)(4 158)(5 159)(6 160)(7 141)(8 142)(9 143)(10 144)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 121)(40 122)(41 118)(42 119)(43 120)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)
(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 77 109 25)(2 76 110 24)(3 75 111 23)(4 74 112 22)(5 73 113 21)(6 72 114 40)(7 71 115 39)(8 70 116 38)(9 69 117 37)(10 68 118 36)(11 67 119 35)(12 66 120 34)(13 65 101 33)(14 64 102 32)(15 63 103 31)(16 62 104 30)(17 61 105 29)(18 80 106 28)(19 79 107 27)(20 78 108 26)(41 138 144 87)(42 137 145 86)(43 136 146 85)(44 135 147 84)(45 134 148 83)(46 133 149 82)(47 132 150 81)(48 131 151 100)(49 130 152 99)(50 129 153 98)(51 128 154 97)(52 127 155 96)(53 126 156 95)(54 125 157 94)(55 124 158 93)(56 123 159 92)(57 122 160 91)(58 121 141 90)(59 140 142 89)(60 139 143 88)
(1 42)(2 41)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 121)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(61 82)(62 81)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(101 153)(102 152)(103 151)(104 150)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 143)(112 142)(113 141)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 154)

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I 4J 4K 4L 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 20 20 20 20 2 ··· 2 20 20 20 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 C5⋊D4 D20 C4○D20 kernel C2×C20⋊7D4 C2×C4⋊Dic5 C2×D10⋊C4 C20⋊7D4 C22×D20 C22×C5⋊D4 C23×C20 C2×C20 C22×C10 C23×C4 C2×C10 C22×C4 C24 C2×C4 C23 C22 # reps 1 1 2 8 1 2 1 4 4 2 4 12 2 16 16 16

Matrix representation of C2×C207D4 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 40 8 0 0 0 40 7 0 0 0 0 0 13 2 0 0 0 39 25
,
 40 0 0 0 0 0 38 20 0 0 0 20 3 0 0 0 0 0 40 0 0 0 0 35 1
,
 1 0 0 0 0 0 7 33 0 0 0 6 34 0 0 0 0 0 40 0 0 0 0 35 1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,40,0,0,0,8,7,0,0,0,0,0,13,39,0,0,0,2,25],[40,0,0,0,0,0,38,20,0,0,0,20,3,0,0,0,0,0,40,35,0,0,0,0,1],[1,0,0,0,0,0,7,6,0,0,0,33,34,0,0,0,0,0,40,35,0,0,0,0,1] >;

C2×C207D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes_7D_4
% in TeX

G:=Group("C2xC20:7D4");
// GroupNames label

G:=SmallGroup(320,1462);
// by ID

G=gap.SmallGroup(320,1462);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,12550]);
// Polycyclic

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

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