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## G = C2×C13⋊M4(2)  order 416 = 25·13

### Direct product of C2 and C13⋊M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×C13⋊M4(2)
 Chief series C1 — C13 — C26 — Dic13 — C13⋊C8 — C2×C13⋊C8 — C2×C13⋊M4(2)
 Lower central C13 — C26 — C2×C13⋊M4(2)
 Upper central C1 — C22 — C23

Generators and relations for C2×C13⋊M4(2)
G = < a,b,c,d | a2=b13=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, bd=db, dcd=c5 >

Subgroups: 340 in 68 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C23, C13, C2×C8, M4(2), C22×C4, C26, C26, C26, C2×M4(2), Dic13, Dic13, C2×C26, C2×C26, C2×C26, C13⋊C8, C2×Dic13, C2×Dic13, C22×C26, C2×C13⋊C8, C13⋊M4(2), C22×Dic13, C2×C13⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C2×M4(2), C13⋊C4, C2×C13⋊C4, C13⋊M4(2), C22×C13⋊C4, C2×C13⋊M4(2)

Smallest permutation representation of C2×C13⋊M4(2)
On 208 points
Generators in S208
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 79)(38 80)(39 81)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(105 158)(106 159)(107 160)(108 161)(109 162)(110 163)(111 164)(112 165)(113 166)(114 167)(115 168)(116 169)(117 157)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 181)(128 182)(129 170)(130 171)(131 186)(132 187)(133 188)(134 189)(135 190)(136 191)(137 192)(138 193)(139 194)(140 195)(141 183)(142 184)(143 185)(144 198)(145 199)(146 200)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(153 207)(154 208)(155 196)(156 197)
(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)
(1 194 35 175 14 199 41 157)(2 189 34 180 15 207 40 162)(3 184 33 172 16 202 52 167)(4 192 32 177 17 197 51 159)(5 187 31 182 18 205 50 164)(6 195 30 174 19 200 49 169)(7 190 29 179 20 208 48 161)(8 185 28 171 21 203 47 166)(9 193 27 176 22 198 46 158)(10 188 39 181 23 206 45 163)(11 183 38 173 24 201 44 168)(12 191 37 178 25 196 43 160)(13 186 36 170 26 204 42 165)(53 139 90 121 66 145 100 117)(54 134 89 126 67 153 99 109)(55 142 88 118 68 148 98 114)(56 137 87 123 69 156 97 106)(57 132 86 128 70 151 96 111)(58 140 85 120 71 146 95 116)(59 135 84 125 72 154 94 108)(60 143 83 130 73 149 93 113)(61 138 82 122 74 144 92 105)(62 133 81 127 75 152 104 110)(63 141 80 119 76 147 103 115)(64 136 79 124 77 155 102 107)(65 131 91 129 78 150 101 112)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 79)(38 80)(39 81)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(105 176)(106 177)(107 178)(108 179)(109 180)(110 181)(111 182)(112 170)(113 171)(114 172)(115 173)(116 174)(117 175)(118 167)(119 168)(120 169)(121 157)(122 158)(123 159)(124 160)(125 161)(126 162)(127 163)(128 164)(129 165)(130 166)(131 204)(132 205)(133 206)(134 207)(135 208)(136 196)(137 197)(138 198)(139 199)(140 200)(141 201)(142 202)(143 203)(144 193)(145 194)(146 195)(147 183)(148 184)(149 185)(150 186)(151 187)(152 188)(153 189)(154 190)(155 191)(156 192)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A ··· 8H 13A 13B 13C 26A ··· 26U order 1 2 2 2 2 2 4 4 4 4 4 4 8 ··· 8 13 13 13 26 ··· 26 size 1 1 1 1 2 2 13 13 13 13 26 26 26 ··· 26 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 M4(2) C13⋊C4 C2×C13⋊C4 C13⋊M4(2) kernel C2×C13⋊M4(2) C2×C13⋊C8 C13⋊M4(2) C22×Dic13 C2×Dic13 C22×C26 C26 C23 C22 C2 # reps 1 2 4 1 6 2 4 3 9 12

Matrix representation of C2×C13⋊M4(2) in GL6(𝔽313)

 312 0 0 0 0 0 0 312 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 86 312 0 0 0 0 87 312 0 0 0 0 177 86 151 286 0 0 81 135 27 47
,
 312 173 0 0 0 0 286 1 0 0 0 0 0 0 0 0 312 1 0 0 247 288 311 85 0 0 311 173 25 0 0 0 254 141 25 0
,
 312 116 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 209 27 312 0 0 0 209 27 0 312

G:=sub<GL(6,GF(313))| [312,0,0,0,0,0,0,312,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,86,87,177,81,0,0,312,312,86,135,0,0,0,0,151,27,0,0,0,0,286,47],[312,286,0,0,0,0,173,1,0,0,0,0,0,0,0,247,311,254,0,0,0,288,173,141,0,0,312,311,25,25,0,0,1,85,0,0],[312,0,0,0,0,0,116,1,0,0,0,0,0,0,1,0,209,209,0,0,0,1,27,27,0,0,0,0,312,0,0,0,0,0,0,312] >;

C2×C13⋊M4(2) in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes M_4(2)
% in TeX

G:=Group("C2xC13:M4(2)");
// GroupNames label

G:=SmallGroup(416,210);
// by ID

G=gap.SmallGroup(416,210);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,69,9221,1751]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^13=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,b*d=d*b,d*c*d=c^5>;
// generators/relations

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