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## G = C42⋊2D13order 416 = 25·13

### 2nd semidirect product of C42 and D13 acting via D13/C13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C42⋊2D13
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — D26⋊C4 — C42⋊2D13
 Lower central C13 — C2×C26 — C42⋊2D13
 Upper central C1 — C22 — C42

Generators and relations for C422D13
G = < a,b,c,d | a4=b4=c13=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >

Subgroups: 408 in 60 conjugacy classes, 29 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C13, C42, C22⋊C4, C4⋊C4, D13, C26, C422C2, Dic13, C52, D26, C2×C26, C2×Dic13, C2×C52, C22×D13, C26.D4, D26⋊C4, C4×C52, C422D13
Quotients: C1, C2, C22, C23, C4○D4, D13, C422C2, D26, C22×D13, D525C2, C422D13

Smallest permutation representation of C422D13
On 208 points
Generators in S208
```(1 134 29 115)(2 135 30 116)(3 136 31 117)(4 137 32 105)(5 138 33 106)(6 139 34 107)(7 140 35 108)(8 141 36 109)(9 142 37 110)(10 143 38 111)(11 131 39 112)(12 132 27 113)(13 133 28 114)(14 156 49 127)(15 144 50 128)(16 145 51 129)(17 146 52 130)(18 147 40 118)(19 148 41 119)(20 149 42 120)(21 150 43 121)(22 151 44 122)(23 152 45 123)(24 153 46 124)(25 154 47 125)(26 155 48 126)(53 189 86 164)(54 190 87 165)(55 191 88 166)(56 192 89 167)(57 193 90 168)(58 194 91 169)(59 195 79 157)(60 183 80 158)(61 184 81 159)(62 185 82 160)(63 186 83 161)(64 187 84 162)(65 188 85 163)(66 202 96 175)(67 203 97 176)(68 204 98 177)(69 205 99 178)(70 206 100 179)(71 207 101 180)(72 208 102 181)(73 196 103 182)(74 197 104 170)(75 198 92 171)(76 199 93 172)(77 200 94 173)(78 201 95 174)
(1 71 19 63)(2 72 20 64)(3 73 21 65)(4 74 22 53)(5 75 23 54)(6 76 24 55)(7 77 25 56)(8 78 26 57)(9 66 14 58)(10 67 15 59)(11 68 16 60)(12 69 17 61)(13 70 18 62)(27 99 52 81)(28 100 40 82)(29 101 41 83)(30 102 42 84)(31 103 43 85)(32 104 44 86)(33 92 45 87)(34 93 46 88)(35 94 47 89)(36 95 48 90)(37 96 49 91)(38 97 50 79)(39 98 51 80)(105 170 122 164)(106 171 123 165)(107 172 124 166)(108 173 125 167)(109 174 126 168)(110 175 127 169)(111 176 128 157)(112 177 129 158)(113 178 130 159)(114 179 118 160)(115 180 119 161)(116 181 120 162)(117 182 121 163)(131 204 145 183)(132 205 146 184)(133 206 147 185)(134 207 148 186)(135 208 149 187)(136 196 150 188)(137 197 151 189)(138 198 152 190)(139 199 153 191)(140 200 154 192)(141 201 155 193)(142 202 156 194)(143 203 144 195)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 23)(15 22)(16 21)(17 20)(18 19)(24 26)(27 30)(28 29)(31 39)(32 38)(33 37)(34 36)(40 41)(42 52)(43 51)(44 50)(45 49)(46 48)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 91)(76 90)(77 89)(78 88)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 130)(117 129)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 156)(139 155)(140 154)(141 153)(142 152)(143 151)(157 189)(158 188)(159 187)(160 186)(161 185)(162 184)(163 183)(164 195)(165 194)(166 193)(167 192)(168 191)(169 190)(170 203)(171 202)(172 201)(173 200)(174 199)(175 198)(176 197)(177 196)(178 208)(179 207)(180 206)(181 205)(182 204)```

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

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

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

110 conjugacy classes

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

110 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4○D4 D13 D26 D52⋊5C2 kernel C42⋊2D13 C26.D4 D26⋊C4 C4×C52 C26 C42 C2×C4 C2 # reps 1 3 3 1 6 6 18 72

Matrix representation of C422D13 in GL4(𝔽53) generated by

 30 0 0 0 0 30 0 0 0 0 17 34 0 0 32 36
,
 5 3 0 0 45 48 0 0 0 0 30 0 0 0 0 30
,
 15 1 0 0 15 47 0 0 0 0 6 1 0 0 29 5
,
 47 52 0 0 35 6 0 0 0 0 6 39 0 0 29 47
`G:=sub<GL(4,GF(53))| [30,0,0,0,0,30,0,0,0,0,17,32,0,0,34,36],[5,45,0,0,3,48,0,0,0,0,30,0,0,0,0,30],[15,15,0,0,1,47,0,0,0,0,6,29,0,0,1,5],[47,35,0,0,52,6,0,0,0,0,6,29,0,0,39,47] >;`

C422D13 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_2D_{13}`
`% in TeX`

`G:=Group("C4^2:2D13");`
`// GroupNames label`

`G:=SmallGroup(416,97);`
`// by ID`

`G=gap.SmallGroup(416,97);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,55,506,86,13829]);`
`// Polycyclic`

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

׿
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