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## G = C4⋊C4×D13order 416 = 25·13

### Direct product of C4⋊C4 and D13

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊C4×D13
G = < a,b,c,d | a4=b4=c13=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 576 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, C23, C13, C4⋊C4, C4⋊C4, C22×C4, D13, C26, C2×C4⋊C4, Dic13, Dic13, C52, C52, D26, C2×C26, C4×D13, C4×D13, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C22×D13, C26.D4, C523C4, C13×C4⋊C4, C2×C4×D13, C2×C4×D13, C4⋊C4×D13
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, D13, C2×C4⋊C4, D26, C4×D13, C22×D13, C2×C4×D13, D4×D13, Q8×D13, C4⋊C4×D13

Smallest permutation representation of C4⋊C4×D13
On 208 points
Generators in S208
(1 151 20 134)(2 152 21 135)(3 153 22 136)(4 154 23 137)(5 155 24 138)(6 156 25 139)(7 144 26 140)(8 145 14 141)(9 146 15 142)(10 147 16 143)(11 148 17 131)(12 149 18 132)(13 150 19 133)(27 128 51 106)(28 129 52 107)(29 130 40 108)(30 118 41 109)(31 119 42 110)(32 120 43 111)(33 121 44 112)(34 122 45 113)(35 123 46 114)(36 124 47 115)(37 125 48 116)(38 126 49 117)(39 127 50 105)(53 200 69 186)(54 201 70 187)(55 202 71 188)(56 203 72 189)(57 204 73 190)(58 205 74 191)(59 206 75 192)(60 207 76 193)(61 208 77 194)(62 196 78 195)(63 197 66 183)(64 198 67 184)(65 199 68 185)(79 173 93 161)(80 174 94 162)(81 175 95 163)(82 176 96 164)(83 177 97 165)(84 178 98 166)(85 179 99 167)(86 180 100 168)(87 181 101 169)(88 182 102 157)(89 170 103 158)(90 171 104 159)(91 172 92 160)
(1 85 29 62)(2 86 30 63)(3 87 31 64)(4 88 32 65)(5 89 33 53)(6 90 34 54)(7 91 35 55)(8 79 36 56)(9 80 37 57)(10 81 38 58)(11 82 39 59)(12 83 27 60)(13 84 28 61)(14 93 47 72)(15 94 48 73)(16 95 49 74)(17 96 50 75)(18 97 51 76)(19 98 52 77)(20 99 40 78)(21 100 41 66)(22 101 42 67)(23 102 43 68)(24 103 44 69)(25 104 45 70)(26 92 46 71)(105 206 131 176)(106 207 132 177)(107 208 133 178)(108 196 134 179)(109 197 135 180)(110 198 136 181)(111 199 137 182)(112 200 138 170)(113 201 139 171)(114 202 140 172)(115 203 141 173)(116 204 142 174)(117 205 143 175)(118 183 152 168)(119 184 153 169)(120 185 154 157)(121 186 155 158)(122 187 156 159)(123 188 144 160)(124 189 145 161)(125 190 146 162)(126 191 147 163)(127 192 148 164)(128 193 149 165)(129 194 150 166)(130 195 151 167)
(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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 39)(23 38)(24 37)(25 36)(26 35)(53 94)(54 93)(55 92)(56 104)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 83)(67 82)(68 81)(69 80)(70 79)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(105 153)(106 152)(107 151)(108 150)(109 149)(110 148)(111 147)(112 146)(113 145)(114 144)(115 156)(116 155)(117 154)(118 132)(119 131)(120 143)(121 142)(122 141)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(157 205)(158 204)(159 203)(160 202)(161 201)(162 200)(163 199)(164 198)(165 197)(166 196)(167 208)(168 207)(169 206)(170 190)(171 189)(172 188)(173 187)(174 186)(175 185)(176 184)(177 183)(178 195)(179 194)(180 193)(181 192)(182 191)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 4G ··· 4L 13A ··· 13F 26A ··· 26R 52A ··· 52AJ 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 2 ··· 2 26 ··· 26 2 ··· 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 D4 Q8 D13 D26 C4×D13 D4×D13 Q8×D13 kernel C4⋊C4×D13 C26.D4 C52⋊3C4 C13×C4⋊C4 C2×C4×D13 C4×D13 D26 D26 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 2 1 1 3 8 2 2 6 18 24 6 6

Matrix representation of C4⋊C4×D13 in GL4(𝔽53) generated by

 1 0 0 0 0 1 0 0 0 0 37 9 0 0 48 16
,
 23 0 0 0 0 23 0 0 0 0 52 17 0 0 0 1
,
 0 1 0 0 52 26 0 0 0 0 1 0 0 0 0 1
,
 0 52 0 0 52 0 0 0 0 0 52 0 0 0 0 52
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,37,48,0,0,9,16],[23,0,0,0,0,23,0,0,0,0,52,0,0,0,17,1],[0,52,0,0,1,26,0,0,0,0,1,0,0,0,0,1],[0,52,0,0,52,0,0,0,0,0,52,0,0,0,0,52] >;

C4⋊C4×D13 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times D_{13}
% in TeX

G:=Group("C4:C4xD13");
// GroupNames label

G:=SmallGroup(416,112);
// by ID

G=gap.SmallGroup(416,112);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,188,50,13829]);
// Polycyclic

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

׿
×
𝔽