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## G = C4⋊C4⋊D13order 416 = 25·13

### 6th semidirect product of C4⋊C4 and D13 acting via D13/C13=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C4⋊C4⋊D13
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — D26⋊C4 — C4⋊C4⋊D13
 Lower central C13 — C2×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, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 408 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C13, C42, C22⋊C4, C4⋊C4, C4⋊C4, D13, C26, C422C2, Dic13, C52, D26, C2×C26, C2×Dic13, C2×C52, C22×D13, C4×Dic13, C26.D4, C523C4, D26⋊C4, C13×C4⋊C4, C4⋊C4⋊D13
Quotients: C1, C2, C22, C23, C4○D4, D13, C422C2, D26, C22×D13, D525C2, D42D13, D52⋊C2, C4⋊C4⋊D13

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4○D4 D13 D26 D52⋊5C2 D4⋊2D13 D52⋊C2 kernel C4⋊C4⋊D13 C4×Dic13 C26.D4 C52⋊3C4 D26⋊C4 C13×C4⋊C4 C26 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 1 1 1 3 1 6 6 18 24 6 6

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

 30 0 0 0 0 23 0 0 0 0 42 18 0 0 11 11
,
 0 1 0 0 1 0 0 0 0 0 23 0 0 0 0 23
,
 1 0 0 0 0 1 0 0 0 0 33 1 0 0 33 46
,
 1 0 0 0 0 52 0 0 0 0 12 34 0 0 41 41
`G:=sub<GL(4,GF(53))| [30,0,0,0,0,23,0,0,0,0,42,11,0,0,18,11],[0,1,0,0,1,0,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,1,0,0,0,0,33,33,0,0,1,46],[1,0,0,0,0,52,0,0,0,0,12,41,0,0,34,41] >;`

C4⋊C4⋊D13 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(416,119);`
`// by ID`

`G=gap.SmallGroup(416,119);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,55,218,188,86,13829]);`
`// 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,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

׿
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