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## G = C4×C13⋊D4order 416 = 25·13

### Direct product of C4 and C13⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C4×C13⋊D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×C13⋊D4 — C4×C13⋊D4
 Lower central C13 — C26 — C4×C13⋊D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C4×C13⋊D4
G = < a,b,c,d | a4=b13=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 544 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, D13, C26, C26, C4×D4, Dic13, Dic13, C52, C52, D26, D26, C2×C26, C2×C26, C2×C26, C4×D13, C2×Dic13, C13⋊D4, C2×C52, C2×C52, C22×D13, C22×C26, C4×Dic13, C26.D4, D26⋊C4, C23.D13, C2×C4×D13, C2×C13⋊D4, C22×C52, C4×C13⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, D26, C4×D13, C13⋊D4, C22×D13, C2×C4×D13, D525C2, C2×C13⋊D4, C4×C13⋊D4

Smallest permutation representation of C4×C13⋊D4
On 208 points
Generators in S208
(1 79 27 53)(2 80 28 54)(3 81 29 55)(4 82 30 56)(5 83 31 57)(6 84 32 58)(7 85 33 59)(8 86 34 60)(9 87 35 61)(10 88 36 62)(11 89 37 63)(12 90 38 64)(13 91 39 65)(14 92 40 66)(15 93 41 67)(16 94 42 68)(17 95 43 69)(18 96 44 70)(19 97 45 71)(20 98 46 72)(21 99 47 73)(22 100 48 74)(23 101 49 75)(24 102 50 76)(25 103 51 77)(26 104 52 78)(105 183 131 157)(106 184 132 158)(107 185 133 159)(108 186 134 160)(109 187 135 161)(110 188 136 162)(111 189 137 163)(112 190 138 164)(113 191 139 165)(114 192 140 166)(115 193 141 167)(116 194 142 168)(117 195 143 169)(118 196 144 170)(119 197 145 171)(120 198 146 172)(121 199 147 173)(122 200 148 174)(123 201 149 175)(124 202 150 176)(125 203 151 177)(126 204 152 178)(127 205 153 179)(128 206 154 180)(129 207 155 181)(130 208 156 182)
(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)
(1 105 14 118)(2 117 15 130)(3 116 16 129)(4 115 17 128)(5 114 18 127)(6 113 19 126)(7 112 20 125)(8 111 21 124)(9 110 22 123)(10 109 23 122)(11 108 24 121)(12 107 25 120)(13 106 26 119)(27 131 40 144)(28 143 41 156)(29 142 42 155)(30 141 43 154)(31 140 44 153)(32 139 45 152)(33 138 46 151)(34 137 47 150)(35 136 48 149)(36 135 49 148)(37 134 50 147)(38 133 51 146)(39 132 52 145)(53 157 66 170)(54 169 67 182)(55 168 68 181)(56 167 69 180)(57 166 70 179)(58 165 71 178)(59 164 72 177)(60 163 73 176)(61 162 74 175)(62 161 75 174)(63 160 76 173)(64 159 77 172)(65 158 78 171)(79 183 92 196)(80 195 93 208)(81 194 94 207)(82 193 95 206)(83 192 96 205)(84 191 97 204)(85 190 98 203)(86 189 99 202)(87 188 100 201)(88 187 101 200)(89 186 102 199)(90 185 103 198)(91 184 104 197)
(1 27)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 40)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(53 79)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 92)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(105 144)(106 156)(107 155)(108 154)(109 153)(110 152)(111 151)(112 150)(113 149)(114 148)(115 147)(116 146)(117 145)(118 131)(119 143)(120 142)(121 141)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(157 196)(158 208)(159 207)(160 206)(161 205)(162 204)(163 203)(164 202)(165 201)(166 200)(167 199)(168 198)(169 197)(170 183)(171 195)(172 194)(173 193)(174 192)(175 191)(176 190)(177 189)(178 188)(179 187)(180 186)(181 185)(182 184)

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

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

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

116 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G ··· 4L 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 26 26 1 1 1 1 2 2 26 ··· 26 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 C4○D4 D13 D26 D26 C13⋊D4 C4×D13 D52⋊5C2 kernel C4×C13⋊D4 C4×Dic13 C26.D4 D26⋊C4 C23.D13 C2×C4×D13 C2×C13⋊D4 C22×C52 C13⋊D4 C52 C26 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 6 12 6 24 24 24

Matrix representation of C4×C13⋊D4 in GL3(𝔽53) generated by

 23 0 0 0 1 0 0 0 1
,
 1 0 0 0 19 21 0 1 43
,
 1 0 0 0 33 19 0 4 20
,
 1 0 0 0 18 46 0 31 35
G:=sub<GL(3,GF(53))| [23,0,0,0,1,0,0,0,1],[1,0,0,0,19,1,0,21,43],[1,0,0,0,33,4,0,19,20],[1,0,0,0,18,31,0,46,35] >;

C4×C13⋊D4 in GAP, Magma, Sage, TeX

C_4\times C_{13}\rtimes D_4
% in TeX

G:=Group("C4xC13:D4");
// GroupNames label

G:=SmallGroup(416,149);
// by ID

G=gap.SmallGroup(416,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,50,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^13=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

׿
×
𝔽