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## G = C42×D13order 416 = 25·13

### Direct product of C42 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C42×D13
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C4×D13 — C42×D13
 Lower central C13 — C42×D13
 Upper central C1 — C42

Generators and relations for C42×D13
G = < a,b,c,d | a4=b4=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 576 in 108 conjugacy classes, 69 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C4 [×6], C22, C22 [×6], C2×C4 [×3], C2×C4 [×15], C23, C13, C42, C42 [×3], C22×C4 [×3], D13 [×4], C26 [×3], C2×C42, Dic13 [×6], C52 [×6], D26 [×6], C2×C26, C4×D13 [×12], C2×Dic13 [×3], C2×C52 [×3], C22×D13, C4×Dic13 [×3], C4×C52, C2×C4×D13 [×3], C42×D13
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], D13, C2×C42, D26 [×3], C4×D13 [×6], C22×D13, C2×C4×D13 [×3], C42×D13

Smallest permutation representation of C42×D13
On 208 points
Generators in S208
(1 119 16 107)(2 120 17 108)(3 121 18 109)(4 122 19 110)(5 123 20 111)(6 124 21 112)(7 125 22 113)(8 126 23 114)(9 127 24 115)(10 128 25 116)(11 129 26 117)(12 130 14 105)(13 118 15 106)(27 149 46 136)(28 150 47 137)(29 151 48 138)(30 152 49 139)(31 153 50 140)(32 154 51 141)(33 155 52 142)(34 156 40 143)(35 144 41 131)(36 145 42 132)(37 146 43 133)(38 147 44 134)(39 148 45 135)(53 175 72 167)(54 176 73 168)(55 177 74 169)(56 178 75 157)(57 179 76 158)(58 180 77 159)(59 181 78 160)(60 182 66 161)(61 170 67 162)(62 171 68 163)(63 172 69 164)(64 173 70 165)(65 174 71 166)(79 199 92 185)(80 200 93 186)(81 201 94 187)(82 202 95 188)(83 203 96 189)(84 204 97 190)(85 205 98 191)(86 206 99 192)(87 207 100 193)(88 208 101 194)(89 196 102 195)(90 197 103 183)(91 198 104 184)
(1 97 32 76)(2 98 33 77)(3 99 34 78)(4 100 35 66)(5 101 36 67)(6 102 37 68)(7 103 38 69)(8 104 39 70)(9 92 27 71)(10 93 28 72)(11 94 29 73)(12 95 30 74)(13 96 31 75)(14 82 49 55)(15 83 50 56)(16 84 51 57)(17 85 52 58)(18 86 40 59)(19 87 41 60)(20 88 42 61)(21 89 43 62)(22 90 44 63)(23 91 45 64)(24 79 46 65)(25 80 47 53)(26 81 48 54)(105 202 139 177)(106 203 140 178)(107 204 141 179)(108 205 142 180)(109 206 143 181)(110 207 131 182)(111 208 132 170)(112 196 133 171)(113 197 134 172)(114 198 135 173)(115 199 136 174)(116 200 137 175)(117 201 138 176)(118 189 153 157)(119 190 154 158)(120 191 155 159)(121 192 156 160)(122 193 144 161)(123 194 145 162)(124 195 146 163)(125 183 147 164)(126 184 148 165)(127 185 149 166)(128 186 150 167)(129 187 151 168)(130 188 152 169)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(53 87)(54 86)(55 85)(56 84)(57 83)(58 82)(59 81)(60 80)(61 79)(62 91)(63 90)(64 89)(65 88)(66 93)(67 92)(68 104)(69 103)(70 102)(71 101)(72 100)(73 99)(74 98)(75 97)(76 96)(77 95)(78 94)(105 142)(106 141)(107 140)(108 139)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 143)(118 154)(119 153)(120 152)(121 151)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 156)(130 155)(157 190)(158 189)(159 188)(160 187)(161 186)(162 185)(163 184)(164 183)(165 195)(166 194)(167 193)(168 192)(169 191)(170 199)(171 198)(172 197)(173 196)(174 208)(175 207)(176 206)(177 205)(178 204)(179 203)(180 202)(181 201)(182 200)

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

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

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

128 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4X 13A ··· 13F 26A ··· 26R 52A ··· 52BT order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 13 13 13 13 1 ··· 1 13 ··· 13 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D13 D26 C4×D13 kernel C42×D13 C4×Dic13 C4×C52 C2×C4×D13 C4×D13 C42 C2×C4 C4 # reps 1 3 1 3 24 6 18 72

Matrix representation of C42×D13 in GL4(𝔽53) generated by

 30 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 52 0 0 0 0 30 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 27 1 0 0 31 11
,
 52 0 0 0 0 1 0 0 0 0 11 52 0 0 14 42
G:=sub<GL(4,GF(53))| [30,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,31,0,0,1,11],[52,0,0,0,0,1,0,0,0,0,11,14,0,0,52,42] >;

C42×D13 in GAP, Magma, Sage, TeX

C_4^2\times D_{13}
% in TeX

G:=Group("C4^2xD13");
// GroupNames label

G:=SmallGroup(416,92);
// by ID

G=gap.SmallGroup(416,92);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,50,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽