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G = C2×D52⋊C2order 416 = 25·13

Direct product of C2 and D52⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×D52⋊C2
 Chief series C1 — C13 — C26 — D26 — C22×D13 — C2×C4×D13 — C2×D52⋊C2
 Lower central C13 — C26 — C2×D52⋊C2
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×D52⋊C2
G = < a,b,c,d | a2=b52=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b25, dcd=b50c >

Subgroups: 1088 in 164 conjugacy classes, 89 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C13, C22×C4, C2×D4, C2×Q8, C4○D4, D13, C26, C26, C2×C4○D4, Dic13, C52, D26, D26, C2×C26, C4×D13, D52, C2×Dic13, C2×C52, Q8×C13, C22×D13, C2×C4×D13, C2×D52, D52⋊C2, Q8×C26, C2×D52⋊C2
Quotients: C1, C2, C22, C23, C4○D4, C24, D13, C2×C4○D4, D26, C22×D13, D52⋊C2, C23×D13, C2×D52⋊C2

Smallest permutation representation of C2×D52⋊C2
On 208 points
Generators in S208
(1 137)(2 138)(3 139)(4 140)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(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)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 167)(54 168)(55 169)(56 170)(57 171)(58 172)(59 173)(60 174)(61 175)(62 176)(63 177)(64 178)(65 179)(66 180)(67 181)(68 182)(69 183)(70 184)(71 185)(72 186)(73 187)(74 188)(75 189)(76 190)(77 191)(78 192)(79 193)(80 194)(81 195)(82 196)(83 197)(84 198)(85 199)(86 200)(87 201)(88 202)(89 203)(90 204)(91 205)(92 206)(93 207)(94 208)(95 157)(96 158)(97 159)(98 160)(99 161)(100 162)(101 163)(102 164)(103 165)(104 166)
(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)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 53)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 84)(23 83)(24 82)(25 81)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 72)(35 71)(36 70)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(105 199)(106 198)(107 197)(108 196)(109 195)(110 194)(111 193)(112 192)(113 191)(114 190)(115 189)(116 188)(117 187)(118 186)(119 185)(120 184)(121 183)(122 182)(123 181)(124 180)(125 179)(126 178)(127 177)(128 176)(129 175)(130 174)(131 173)(132 172)(133 171)(134 170)(135 169)(136 168)(137 167)(138 166)(139 165)(140 164)(141 163)(142 162)(143 161)(144 160)(145 159)(146 158)(147 157)(148 208)(149 207)(150 206)(151 205)(152 204)(153 203)(154 202)(155 201)(156 200)
(2 26)(3 51)(4 24)(5 49)(6 22)(7 47)(8 20)(9 45)(10 18)(11 43)(12 16)(13 41)(15 39)(17 37)(19 35)(21 33)(23 31)(25 29)(28 52)(30 50)(32 48)(34 46)(36 44)(38 42)(53 55)(54 80)(56 78)(57 103)(58 76)(59 101)(60 74)(61 99)(62 72)(63 97)(64 70)(65 95)(66 68)(67 93)(69 91)(71 89)(73 87)(75 85)(77 83)(79 81)(82 104)(84 102)(86 100)(88 98)(90 96)(92 94)(105 117)(106 142)(107 115)(108 140)(109 113)(110 138)(112 136)(114 134)(116 132)(118 130)(119 155)(120 128)(121 153)(122 126)(123 151)(125 149)(127 147)(129 145)(131 143)(133 141)(135 139)(144 156)(146 154)(148 152)(157 179)(158 204)(159 177)(160 202)(161 175)(162 200)(163 173)(164 198)(165 171)(166 196)(167 169)(168 194)(170 192)(172 190)(174 188)(176 186)(178 184)(180 182)(181 207)(183 205)(185 203)(187 201)(189 199)(191 197)(193 195)(206 208)

G:=sub<Sym(208)| (1,137)(2,138)(3,139)(4,140)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(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)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,177)(64,178)(65,179)(66,180)(67,181)(68,182)(69,183)(70,184)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,193)(80,194)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,201)(88,202)(89,203)(90,204)(91,205)(92,206)(93,207)(94,208)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166), (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,53)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(105,199)(106,198)(107,197)(108,196)(109,195)(110,194)(111,193)(112,192)(113,191)(114,190)(115,189)(116,188)(117,187)(118,186)(119,185)(120,184)(121,183)(122,182)(123,181)(124,180)(125,179)(126,178)(127,177)(128,176)(129,175)(130,174)(131,173)(132,172)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)(139,165)(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)(147,157)(148,208)(149,207)(150,206)(151,205)(152,204)(153,203)(154,202)(155,201)(156,200), (2,26)(3,51)(4,24)(5,49)(6,22)(7,47)(8,20)(9,45)(10,18)(11,43)(12,16)(13,41)(15,39)(17,37)(19,35)(21,33)(23,31)(25,29)(28,52)(30,50)(32,48)(34,46)(36,44)(38,42)(53,55)(54,80)(56,78)(57,103)(58,76)(59,101)(60,74)(61,99)(62,72)(63,97)(64,70)(65,95)(66,68)(67,93)(69,91)(71,89)(73,87)(75,85)(77,83)(79,81)(82,104)(84,102)(86,100)(88,98)(90,96)(92,94)(105,117)(106,142)(107,115)(108,140)(109,113)(110,138)(112,136)(114,134)(116,132)(118,130)(119,155)(120,128)(121,153)(122,126)(123,151)(125,149)(127,147)(129,145)(131,143)(133,141)(135,139)(144,156)(146,154)(148,152)(157,179)(158,204)(159,177)(160,202)(161,175)(162,200)(163,173)(164,198)(165,171)(166,196)(167,169)(168,194)(170,192)(172,190)(174,188)(176,186)(178,184)(180,182)(181,207)(183,205)(185,203)(187,201)(189,199)(191,197)(193,195)(206,208)>;

G:=Group( (1,137)(2,138)(3,139)(4,140)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(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)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,167)(54,168)(55,169)(56,170)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,177)(64,178)(65,179)(66,180)(67,181)(68,182)(69,183)(70,184)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,193)(80,194)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,201)(88,202)(89,203)(90,204)(91,205)(92,206)(93,207)(94,208)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166), (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,53)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,54)(105,199)(106,198)(107,197)(108,196)(109,195)(110,194)(111,193)(112,192)(113,191)(114,190)(115,189)(116,188)(117,187)(118,186)(119,185)(120,184)(121,183)(122,182)(123,181)(124,180)(125,179)(126,178)(127,177)(128,176)(129,175)(130,174)(131,173)(132,172)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)(139,165)(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)(147,157)(148,208)(149,207)(150,206)(151,205)(152,204)(153,203)(154,202)(155,201)(156,200), (2,26)(3,51)(4,24)(5,49)(6,22)(7,47)(8,20)(9,45)(10,18)(11,43)(12,16)(13,41)(15,39)(17,37)(19,35)(21,33)(23,31)(25,29)(28,52)(30,50)(32,48)(34,46)(36,44)(38,42)(53,55)(54,80)(56,78)(57,103)(58,76)(59,101)(60,74)(61,99)(62,72)(63,97)(64,70)(65,95)(66,68)(67,93)(69,91)(71,89)(73,87)(75,85)(77,83)(79,81)(82,104)(84,102)(86,100)(88,98)(90,96)(92,94)(105,117)(106,142)(107,115)(108,140)(109,113)(110,138)(112,136)(114,134)(116,132)(118,130)(119,155)(120,128)(121,153)(122,126)(123,151)(125,149)(127,147)(129,145)(131,143)(133,141)(135,139)(144,156)(146,154)(148,152)(157,179)(158,204)(159,177)(160,202)(161,175)(162,200)(163,173)(164,198)(165,171)(166,196)(167,169)(168,194)(170,192)(172,190)(174,188)(176,186)(178,184)(180,182)(181,207)(183,205)(185,203)(187,201)(189,199)(191,197)(193,195)(206,208) );

G=PermutationGroup([[(1,137),(2,138),(3,139),(4,140),(5,141),(6,142),(7,143),(8,144),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,151),(16,152),(17,153),(18,154),(19,155),(20,156),(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),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,167),(54,168),(55,169),(56,170),(57,171),(58,172),(59,173),(60,174),(61,175),(62,176),(63,177),(64,178),(65,179),(66,180),(67,181),(68,182),(69,183),(70,184),(71,185),(72,186),(73,187),(74,188),(75,189),(76,190),(77,191),(78,192),(79,193),(80,194),(81,195),(82,196),(83,197),(84,198),(85,199),(86,200),(87,201),(88,202),(89,203),(90,204),(91,205),(92,206),(93,207),(94,208),(95,157),(96,158),(97,159),(98,160),(99,161),(100,162),(101,163),(102,164),(103,165),(104,166)], [(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),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,53),(2,104),(3,103),(4,102),(5,101),(6,100),(7,99),(8,98),(9,97),(10,96),(11,95),(12,94),(13,93),(14,92),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,84),(23,83),(24,82),(25,81),(26,80),(27,79),(28,78),(29,77),(30,76),(31,75),(32,74),(33,73),(34,72),(35,71),(36,70),(37,69),(38,68),(39,67),(40,66),(41,65),(42,64),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,54),(105,199),(106,198),(107,197),(108,196),(109,195),(110,194),(111,193),(112,192),(113,191),(114,190),(115,189),(116,188),(117,187),(118,186),(119,185),(120,184),(121,183),(122,182),(123,181),(124,180),(125,179),(126,178),(127,177),(128,176),(129,175),(130,174),(131,173),(132,172),(133,171),(134,170),(135,169),(136,168),(137,167),(138,166),(139,165),(140,164),(141,163),(142,162),(143,161),(144,160),(145,159),(146,158),(147,157),(148,208),(149,207),(150,206),(151,205),(152,204),(153,203),(154,202),(155,201),(156,200)], [(2,26),(3,51),(4,24),(5,49),(6,22),(7,47),(8,20),(9,45),(10,18),(11,43),(12,16),(13,41),(15,39),(17,37),(19,35),(21,33),(23,31),(25,29),(28,52),(30,50),(32,48),(34,46),(36,44),(38,42),(53,55),(54,80),(56,78),(57,103),(58,76),(59,101),(60,74),(61,99),(62,72),(63,97),(64,70),(65,95),(66,68),(67,93),(69,91),(71,89),(73,87),(75,85),(77,83),(79,81),(82,104),(84,102),(86,100),(88,98),(90,96),(92,94),(105,117),(106,142),(107,115),(108,140),(109,113),(110,138),(112,136),(114,134),(116,132),(118,130),(119,155),(120,128),(121,153),(122,126),(123,151),(125,149),(127,147),(129,145),(131,143),(133,141),(135,139),(144,156),(146,154),(148,152),(157,179),(158,204),(159,177),(160,202),(161,175),(162,200),(163,173),(164,198),(165,171),(166,196),(167,169),(168,194),(170,192),(172,190),(174,188),(176,186),(178,184),(180,182),(181,207),(183,205),(185,203),(187,201),(189,199),(191,197),(193,195),(206,208)]])

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4F 4G 4H 4I 4J 13A ··· 13F 26A ··· 26R 52A ··· 52AJ order 1 2 2 2 2 ··· 2 4 ··· 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 26 ··· 26 2 ··· 2 13 13 13 13 2 ··· 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C4○D4 D13 D26 D26 D52⋊C2 kernel C2×D52⋊C2 C2×C4×D13 C2×D52 D52⋊C2 Q8×C26 C26 C2×Q8 C2×C4 Q8 C2 # reps 1 3 3 8 1 4 6 18 24 12

Matrix representation of C2×D52⋊C2 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 1 0 0 0 0 1
,
 9 9 0 0 0 6 0 0 0 0 30 0 0 0 49 23
,
 36 24 0 0 41 17 0 0 0 0 5 22 0 0 23 48
,
 49 2 0 0 19 4 0 0 0 0 1 0 0 0 14 52
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,9,6,0,0,0,0,30,49,0,0,0,23],[36,41,0,0,24,17,0,0,0,0,5,23,0,0,22,48],[49,19,0,0,2,4,0,0,0,0,1,14,0,0,0,52] >;

C2×D52⋊C2 in GAP, Magma, Sage, TeX

C_2\times D_{52}\rtimes C_2
% in TeX

G:=Group("C2xD52:C2");
// GroupNames label

G:=SmallGroup(416,220);
// by ID

G=gap.SmallGroup(416,220);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,86,579,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^52=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^25,d*c*d=b^50*c>;
// generators/relations

׿
×
𝔽