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G = D4⋊Dic13order 416 = 25·13

1st semidirect product of D4 and Dic13 acting via Dic13/C26=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.7D4, C26.12D8, D41Dic13, C26.6SD16, (D4×C13)⋊4C4, C52.28(C2×C4), (C2×D4).1D13, (D4×C26).1C2, C523C410C2, (C2×C4).39D26, (C2×C26).33D4, C134(D4⋊C4), C2.3(D4⋊D13), C4.1(C2×Dic13), C4.12(C13⋊D4), (C2×C52).16C22, C2.3(D4.D13), C26.24(C22⋊C4), C2.3(C23.D13), C22.17(C13⋊D4), (C2×C132C8)⋊2C2, SmallGroup(416,39)

Series: Derived Chief Lower central Upper central

C1C52 — D4⋊Dic13
C1C13C26C2×C26C2×C52C523C4 — D4⋊Dic13
C13C26C52 — D4⋊Dic13
C1C22C2×C4C2×D4

Generators and relations for D4⋊Dic13
 G = < a,b,c,d | a4=b2=c26=1, d2=c13, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
4C2
2C22
2C22
4C22
4C22
52C4
4C26
4C26
2D4
2C23
26C2×C4
26C8
2C2×C26
2C2×C26
4C2×C26
4Dic13
4C2×C26
13C4⋊C4
13C2×C8
2C132C8
2D4×C13
2C2×Dic13
2C22×C26
13D4⋊C4

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

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D8A8B8C8D13A···13F26A···26R26S···26AP52A···52L
order1222224444888813···1326···2626···2652···52
size111144225252262626262···22···24···44···4

74 irreducible representations

dim1111122222222244
type+++++++++-+-
imageC1C2C2C2C4D4D4D8SD16D13D26Dic13C13⋊D4C13⋊D4D4⋊D13D4.D13
kernelD4⋊Dic13C2×C132C8C523C4D4×C26D4×C13C52C2×C26C26C26C2×D4C2×C4D4C4C22C2C2
# reps1111411226612121266

Matrix representation of D4⋊Dic13 in GL4(𝔽313) generated by

1000
0100
005744
00168256
,
1000
0100
005744
0040256
,
20131200
2082700
003120
000312
,
311700
13131000
0019743
0051116
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,57,168,0,0,44,256],[1,0,0,0,0,1,0,0,0,0,57,40,0,0,44,256],[201,208,0,0,312,27,0,0,0,0,312,0,0,0,0,312],[3,131,0,0,117,310,0,0,0,0,197,51,0,0,43,116] >;

D4⋊Dic13 in GAP, Magma, Sage, TeX

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

G:=Group("D4:Dic13");
// GroupNames label

G:=SmallGroup(416,39);
// by ID

G=gap.SmallGroup(416,39);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,579,297,69,13829]);
// Polycyclic

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

Export

Subgroup lattice of D4⋊Dic13 in TeX

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