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## G = D4×Dic13order 416 = 25·13

### Direct product of D4 and Dic13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D4×Dic13
 Chief series C1 — C13 — C26 — C2×C26 — C2×Dic13 — C22×Dic13 — D4×Dic13
 Lower central C13 — C26 — D4×Dic13
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×Dic13
G = < a,b,c,d | a4=b2=c26=1, d2=c13, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 448 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C13, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C26 [×3], C26 [×4], C4×D4, Dic13 [×2], Dic13 [×3], C52 [×2], C2×C26, C2×C26 [×4], C2×C26 [×4], C2×Dic13 [×2], C2×Dic13 [×2], C2×Dic13 [×4], C2×C52, D4×C13 [×4], C22×C26 [×2], C4×Dic13, C523C4, C23.D13 [×2], C22×Dic13 [×2], D4×C26, D4×Dic13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, Dic13 [×4], D26 [×3], C2×Dic13 [×6], C22×D13, D4×D13, D42D13, C22×Dic13, D4×Dic13

Smallest permutation representation of D4×Dic13
On 208 points
Generators in S208
(1 116 152 164)(2 117 153 165)(3 118 154 166)(4 119 155 167)(5 120 156 168)(6 121 131 169)(7 122 132 170)(8 123 133 171)(9 124 134 172)(10 125 135 173)(11 126 136 174)(12 127 137 175)(13 128 138 176)(14 129 139 177)(15 130 140 178)(16 105 141 179)(17 106 142 180)(18 107 143 181)(19 108 144 182)(20 109 145 157)(21 110 146 158)(22 111 147 159)(23 112 148 160)(24 113 149 161)(25 114 150 162)(26 115 151 163)(27 187 85 71)(28 188 86 72)(29 189 87 73)(30 190 88 74)(31 191 89 75)(32 192 90 76)(33 193 91 77)(34 194 92 78)(35 195 93 53)(36 196 94 54)(37 197 95 55)(38 198 96 56)(39 199 97 57)(40 200 98 58)(41 201 99 59)(42 202 100 60)(43 203 101 61)(44 204 102 62)(45 205 103 63)(46 206 104 64)(47 207 79 65)(48 208 80 66)(49 183 81 67)(50 184 82 68)(51 185 83 69)(52 186 84 70)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(105 166)(106 167)(107 168)(108 169)(109 170)(110 171)(111 172)(112 173)(113 174)(114 175)(115 176)(116 177)(117 178)(118 179)(119 180)(120 181)(121 182)(122 157)(123 158)(124 159)(125 160)(126 161)(127 162)(128 163)(129 164)(130 165)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 154)(142 155)(143 156)(183 196)(184 197)(185 198)(186 199)(187 200)(188 201)(189 202)(190 203)(191 204)(192 205)(193 206)(194 207)(195 208)
(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)
(1 197 14 184)(2 196 15 183)(3 195 16 208)(4 194 17 207)(5 193 18 206)(6 192 19 205)(7 191 20 204)(8 190 21 203)(9 189 22 202)(10 188 23 201)(11 187 24 200)(12 186 25 199)(13 185 26 198)(27 161 40 174)(28 160 41 173)(29 159 42 172)(30 158 43 171)(31 157 44 170)(32 182 45 169)(33 181 46 168)(34 180 47 167)(35 179 48 166)(36 178 49 165)(37 177 50 164)(38 176 51 163)(39 175 52 162)(53 141 66 154)(54 140 67 153)(55 139 68 152)(56 138 69 151)(57 137 70 150)(58 136 71 149)(59 135 72 148)(60 134 73 147)(61 133 74 146)(62 132 75 145)(63 131 76 144)(64 156 77 143)(65 155 78 142)(79 119 92 106)(80 118 93 105)(81 117 94 130)(82 116 95 129)(83 115 96 128)(84 114 97 127)(85 113 98 126)(86 112 99 125)(87 111 100 124)(88 110 101 123)(89 109 102 122)(90 108 103 121)(91 107 104 120)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 D4 C4○D4 D13 D26 Dic13 D26 D4×D13 D4⋊2D13 kernel D4×Dic13 C4×Dic13 C52⋊3C4 C23.D13 C22×Dic13 D4×C26 D4×C13 Dic13 C26 C2×D4 C2×C4 D4 C23 C2 C2 # reps 1 1 1 2 2 1 8 2 2 6 6 24 12 6 6

Matrix representation of D4×Dic13 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 0 1 0 0 52 0
,
 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 1
,
 14 52 0 0 44 31 0 0 0 0 52 0 0 0 0 52
,
 51 20 0 0 13 2 0 0 0 0 30 0 0 0 0 30
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,0,52,0,0,1,0],[52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,1],[14,44,0,0,52,31,0,0,0,0,52,0,0,0,0,52],[51,13,0,0,20,2,0,0,0,0,30,0,0,0,0,30] >;

D4×Dic13 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_{13}
% in TeX

G:=Group("D4xDic13");
// GroupNames label

G:=SmallGroup(416,155);
// by ID

G=gap.SmallGroup(416,155);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,188,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^26=1,d^2=c^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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