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## G = C2×C20⋊D5order 400 = 24·52

### Direct product of C2 and C20⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C2×C20⋊D5
 Chief series C1 — C5 — C52 — C5×C10 — C2×C5⋊D5 — C22×C5⋊D5 — C2×C20⋊D5
 Lower central C52 — C5×C10 — C2×C20⋊D5
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C20⋊D5
G = < a,b,c,d | a2=b20=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1576 in 216 conjugacy classes, 75 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, D5, C10, C2×D4, C20, D10, C2×C10, C52, D20, C2×C20, C22×D5, C5⋊D5, C5×C10, C5×C10, C2×D20, C5×C20, C2×C5⋊D5, C2×C5⋊D5, C102, C20⋊D5, C10×C20, C22×C5⋊D5, C2×C20⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, D20, C22×D5, C5⋊D5, C2×D20, C2×C5⋊D5, C20⋊D5, C22×C5⋊D5, C2×C20⋊D5

Smallest permutation representation of C2×C20⋊D5
On 200 points
Generators in S200
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(81 163)(82 164)(83 165)(84 166)(85 167)(86 168)(87 169)(88 170)(89 171)(90 172)(91 173)(92 174)(93 175)(94 176)(95 177)(96 178)(97 179)(98 180)(99 161)(100 162)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(141 197)(142 198)(143 199)(144 200)(145 181)(146 182)(147 183)(148 184)(149 185)(150 186)(151 187)(152 188)(153 189)(154 190)(155 191)(156 192)(157 193)(158 194)(159 195)(160 196)
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)
(1 35 101 195 168)(2 36 102 196 169)(3 37 103 197 170)(4 38 104 198 171)(5 39 105 199 172)(6 40 106 200 173)(7 21 107 181 174)(8 22 108 182 175)(9 23 109 183 176)(10 24 110 184 177)(11 25 111 185 178)(12 26 112 186 179)(13 27 113 187 180)(14 28 114 188 161)(15 29 115 189 162)(16 30 116 190 163)(17 31 117 191 164)(18 32 118 192 165)(19 33 119 193 166)(20 34 120 194 167)(41 77 129 153 100)(42 78 130 154 81)(43 79 131 155 82)(44 80 132 156 83)(45 61 133 157 84)(46 62 134 158 85)(47 63 135 159 86)(48 64 136 160 87)(49 65 137 141 88)(50 66 138 142 89)(51 67 139 143 90)(52 68 140 144 91)(53 69 121 145 92)(54 70 122 146 93)(55 71 123 147 94)(56 72 124 148 95)(57 73 125 149 96)(58 74 126 150 97)(59 75 127 151 98)(60 76 128 152 99)
(1 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 84)(9 83)(10 82)(11 81)(12 100)(13 99)(14 98)(15 97)(16 96)(17 95)(18 94)(19 93)(20 92)(21 158)(22 157)(23 156)(24 155)(25 154)(26 153)(27 152)(28 151)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 160)(40 159)(41 179)(42 178)(43 177)(44 176)(45 175)(46 174)(47 173)(48 172)(49 171)(50 170)(51 169)(52 168)(53 167)(54 166)(55 165)(56 164)(57 163)(58 162)(59 161)(60 180)(61 182)(62 181)(63 200)(64 199)(65 198)(66 197)(67 196)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 187)(77 186)(78 185)(79 184)(80 183)(101 140)(102 139)(103 138)(104 137)(105 136)(106 135)(107 134)(108 133)(109 132)(110 131)(111 130)(112 129)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)

G:=sub<Sym(200)| (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,161)(100,162)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(141,197)(142,198)(143,199)(144,200)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)(156,192)(157,193)(158,194)(159,195)(160,196), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200), (1,35,101,195,168)(2,36,102,196,169)(3,37,103,197,170)(4,38,104,198,171)(5,39,105,199,172)(6,40,106,200,173)(7,21,107,181,174)(8,22,108,182,175)(9,23,109,183,176)(10,24,110,184,177)(11,25,111,185,178)(12,26,112,186,179)(13,27,113,187,180)(14,28,114,188,161)(15,29,115,189,162)(16,30,116,190,163)(17,31,117,191,164)(18,32,118,192,165)(19,33,119,193,166)(20,34,120,194,167)(41,77,129,153,100)(42,78,130,154,81)(43,79,131,155,82)(44,80,132,156,83)(45,61,133,157,84)(46,62,134,158,85)(47,63,135,159,86)(48,64,136,160,87)(49,65,137,141,88)(50,66,138,142,89)(51,67,139,143,90)(52,68,140,144,91)(53,69,121,145,92)(54,70,122,146,93)(55,71,123,147,94)(56,72,124,148,95)(57,73,125,149,96)(58,74,126,150,97)(59,75,127,151,98)(60,76,128,152,99), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,84)(9,83)(10,82)(11,81)(12,100)(13,99)(14,98)(15,97)(16,96)(17,95)(18,94)(19,93)(20,92)(21,158)(22,157)(23,156)(24,155)(25,154)(26,153)(27,152)(28,151)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,160)(40,159)(41,179)(42,178)(43,177)(44,176)(45,175)(46,174)(47,173)(48,172)(49,171)(50,170)(51,169)(52,168)(53,167)(54,166)(55,165)(56,164)(57,163)(58,162)(59,161)(60,180)(61,182)(62,181)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(101,140)(102,139)(103,138)(104,137)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121)>;

G:=Group( (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,161)(100,162)(101,135)(102,136)(103,137)(104,138)(105,139)(106,140)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(141,197)(142,198)(143,199)(144,200)(145,181)(146,182)(147,183)(148,184)(149,185)(150,186)(151,187)(152,188)(153,189)(154,190)(155,191)(156,192)(157,193)(158,194)(159,195)(160,196), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200), (1,35,101,195,168)(2,36,102,196,169)(3,37,103,197,170)(4,38,104,198,171)(5,39,105,199,172)(6,40,106,200,173)(7,21,107,181,174)(8,22,108,182,175)(9,23,109,183,176)(10,24,110,184,177)(11,25,111,185,178)(12,26,112,186,179)(13,27,113,187,180)(14,28,114,188,161)(15,29,115,189,162)(16,30,116,190,163)(17,31,117,191,164)(18,32,118,192,165)(19,33,119,193,166)(20,34,120,194,167)(41,77,129,153,100)(42,78,130,154,81)(43,79,131,155,82)(44,80,132,156,83)(45,61,133,157,84)(46,62,134,158,85)(47,63,135,159,86)(48,64,136,160,87)(49,65,137,141,88)(50,66,138,142,89)(51,67,139,143,90)(52,68,140,144,91)(53,69,121,145,92)(54,70,122,146,93)(55,71,123,147,94)(56,72,124,148,95)(57,73,125,149,96)(58,74,126,150,97)(59,75,127,151,98)(60,76,128,152,99), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,84)(9,83)(10,82)(11,81)(12,100)(13,99)(14,98)(15,97)(16,96)(17,95)(18,94)(19,93)(20,92)(21,158)(22,157)(23,156)(24,155)(25,154)(26,153)(27,152)(28,151)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,160)(40,159)(41,179)(42,178)(43,177)(44,176)(45,175)(46,174)(47,173)(48,172)(49,171)(50,170)(51,169)(52,168)(53,167)(54,166)(55,165)(56,164)(57,163)(58,162)(59,161)(60,180)(61,182)(62,181)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(101,140)(102,139)(103,138)(104,137)(105,136)(106,135)(107,134)(108,133)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(115,126)(116,125)(117,124)(118,123)(119,122)(120,121) );

G=PermutationGroup([[(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(81,163),(82,164),(83,165),(84,166),(85,167),(86,168),(87,169),(88,170),(89,171),(90,172),(91,173),(92,174),(93,175),(94,176),(95,177),(96,178),(97,179),(98,180),(99,161),(100,162),(101,135),(102,136),(103,137),(104,138),(105,139),(106,140),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(141,197),(142,198),(143,199),(144,200),(145,181),(146,182),(147,183),(148,184),(149,185),(150,186),(151,187),(152,188),(153,189),(154,190),(155,191),(156,192),(157,193),(158,194),(159,195),(160,196)], [(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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)], [(1,35,101,195,168),(2,36,102,196,169),(3,37,103,197,170),(4,38,104,198,171),(5,39,105,199,172),(6,40,106,200,173),(7,21,107,181,174),(8,22,108,182,175),(9,23,109,183,176),(10,24,110,184,177),(11,25,111,185,178),(12,26,112,186,179),(13,27,113,187,180),(14,28,114,188,161),(15,29,115,189,162),(16,30,116,190,163),(17,31,117,191,164),(18,32,118,192,165),(19,33,119,193,166),(20,34,120,194,167),(41,77,129,153,100),(42,78,130,154,81),(43,79,131,155,82),(44,80,132,156,83),(45,61,133,157,84),(46,62,134,158,85),(47,63,135,159,86),(48,64,136,160,87),(49,65,137,141,88),(50,66,138,142,89),(51,67,139,143,90),(52,68,140,144,91),(53,69,121,145,92),(54,70,122,146,93),(55,71,123,147,94),(56,72,124,148,95),(57,73,125,149,96),(58,74,126,150,97),(59,75,127,151,98),(60,76,128,152,99)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,84),(9,83),(10,82),(11,81),(12,100),(13,99),(14,98),(15,97),(16,96),(17,95),(18,94),(19,93),(20,92),(21,158),(22,157),(23,156),(24,155),(25,154),(26,153),(27,152),(28,151),(29,150),(30,149),(31,148),(32,147),(33,146),(34,145),(35,144),(36,143),(37,142),(38,141),(39,160),(40,159),(41,179),(42,178),(43,177),(44,176),(45,175),(46,174),(47,173),(48,172),(49,171),(50,170),(51,169),(52,168),(53,167),(54,166),(55,165),(56,164),(57,163),(58,162),(59,161),(60,180),(61,182),(62,181),(63,200),(64,199),(65,198),(66,197),(67,196),(68,195),(69,194),(70,193),(71,192),(72,191),(73,190),(74,189),(75,188),(76,187),(77,186),(78,185),(79,184),(80,183),(101,140),(102,139),(103,138),(104,137),(105,136),(106,135),(107,134),(108,133),(109,132),(110,131),(111,130),(112,129),(113,128),(114,127),(115,126),(116,125),(117,124),(118,123),(119,122),(120,121)]])

106 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A ··· 5L 10A ··· 10AJ 20A ··· 20AV order 1 2 2 2 2 2 2 2 4 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 1 1 50 50 50 50 2 2 2 ··· 2 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D5 D10 D10 D20 kernel C2×C20⋊D5 C20⋊D5 C10×C20 C22×C5⋊D5 C5×C10 C2×C20 C20 C2×C10 C10 # reps 1 4 1 2 2 12 24 12 48

Matrix representation of C2×C20⋊D5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 32 30 0 0 11 27 0 0 0 0 30 32 0 0 9 11
,
 0 1 0 0 40 34 0 0 0 0 0 1 0 0 40 34
,
 32 30 0 0 11 9 0 0 0 0 30 14 0 0 9 11
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[32,11,0,0,30,27,0,0,0,0,30,9,0,0,32,11],[0,40,0,0,1,34,0,0,0,0,0,40,0,0,1,34],[32,11,0,0,30,9,0,0,0,0,30,9,0,0,14,11] >;

C2×C20⋊D5 in GAP, Magma, Sage, TeX

C_2\times C_{20}\rtimes D_5
% in TeX

G:=Group("C2xC20:D5");
// GroupNames label

G:=SmallGroup(400,193);
// by ID

G=gap.SmallGroup(400,193);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,218,50,1924,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽