Copied to
clipboard

## 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, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C26, C26, C4×D4, Dic13, Dic13, C52, C2×C26, C2×C26, C2×C26, C2×Dic13, C2×Dic13, C2×Dic13, C2×C52, D4×C13, C22×C26, C4×Dic13, C523C4, C23.D13, C22×Dic13, D4×C26, D4×Dic13
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, Dic13, D26, C2×Dic13, C22×D13, D4×D13, D42D13, C22×Dic13, D4×Dic13

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

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

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

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

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

׿
×
𝔽