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G = Q16⋊D13order 416 = 25·13

2nd semidirect product of Q16 and D13 acting via D13/C13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.3D26, Q162D13, Q8.4D26, D26.17D4, C52.9C23, D52.4C22, C104.10C22, Dic13.19D4, Dic26.5C22, Q8⋊D133C2, (Q8×D13)⋊3C2, C104⋊C24C2, C8⋊D134C2, (C13×Q16)⋊4C2, C13⋊Q164C2, C26.35(C2×D4), C2.23(D4×D13), C133(C8.C22), C4.9(C22×D13), D52⋊C2.1C2, C132C8.2C22, (C4×D13).4C22, (Q8×C13).4C22, SmallGroup(416,139)

Series: Derived Chief Lower central Upper central

C1C52 — Q16⋊D13
C1C13C26C52C4×D13Q8×D13 — Q16⋊D13
C13C26C52 — Q16⋊D13
C1C2C4Q16

Generators and relations for Q16⋊D13
 G = < a,b,c,d | a8=c13=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 456 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×4], C22 [×2], C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], C13, M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, D13 [×2], C26, C8.C22, Dic13, Dic13, C52, C52 [×2], D26, D26, C132C8, C104, Dic26, Dic26, C4×D13, C4×D13 [×2], D52, D52, Q8×C13 [×2], C8⋊D13, C104⋊C2, Q8⋊D13, C13⋊Q16, C13×Q16, Q8×D13, D52⋊C2, Q16⋊D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8.C22, D26 [×3], C22×D13, D4×D13, Q16⋊D13

Smallest permutation representation of Q16⋊D13
On 208 points
Generators in S208
(1 193 30 180 22 203 51 160)(2 194 31 181 23 204 52 161)(3 195 32 182 24 205 40 162)(4 183 33 170 25 206 41 163)(5 184 34 171 26 207 42 164)(6 185 35 172 14 208 43 165)(7 186 36 173 15 196 44 166)(8 187 37 174 16 197 45 167)(9 188 38 175 17 198 46 168)(10 189 39 176 18 199 47 169)(11 190 27 177 19 200 48 157)(12 191 28 178 20 201 49 158)(13 192 29 179 21 202 50 159)(53 143 103 107 67 155 90 126)(54 131 104 108 68 156 91 127)(55 132 92 109 69 144 79 128)(56 133 93 110 70 145 80 129)(57 134 94 111 71 146 81 130)(58 135 95 112 72 147 82 118)(59 136 96 113 73 148 83 119)(60 137 97 114 74 149 84 120)(61 138 98 115 75 150 85 121)(62 139 99 116 76 151 86 122)(63 140 100 117 77 152 87 123)(64 141 101 105 78 153 88 124)(65 142 102 106 66 154 89 125)
(1 66 22 65)(2 67 23 53)(3 68 24 54)(4 69 25 55)(5 70 26 56)(6 71 14 57)(7 72 15 58)(8 73 16 59)(9 74 17 60)(10 75 18 61)(11 76 19 62)(12 77 20 63)(13 78 21 64)(27 99 48 86)(28 100 49 87)(29 101 50 88)(30 102 51 89)(31 103 52 90)(32 104 40 91)(33 92 41 79)(34 93 42 80)(35 94 43 81)(36 95 44 82)(37 96 45 83)(38 97 46 84)(39 98 47 85)(105 202 124 192)(106 203 125 193)(107 204 126 194)(108 205 127 195)(109 206 128 183)(110 207 129 184)(111 208 130 185)(112 196 118 186)(113 197 119 187)(114 198 120 188)(115 199 121 189)(116 200 122 190)(117 201 123 191)(131 162 156 182)(132 163 144 170)(133 164 145 171)(134 165 146 172)(135 166 147 173)(136 167 148 174)(137 168 149 175)(138 169 150 176)(139 157 151 177)(140 158 152 178)(141 159 153 179)(142 160 154 180)(143 161 155 181)
(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 16)(17 26)(18 25)(19 24)(20 23)(21 22)(27 32)(28 31)(29 30)(33 39)(34 38)(35 37)(40 48)(41 47)(42 46)(43 45)(49 52)(50 51)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(65 78)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(105 106)(107 117)(108 116)(109 115)(110 114)(111 113)(119 130)(120 129)(121 128)(122 127)(123 126)(124 125)(131 139)(132 138)(133 137)(134 136)(140 143)(141 142)(144 150)(145 149)(146 148)(151 156)(152 155)(153 154)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(183 199)(184 198)(185 197)(186 196)(187 208)(188 207)(189 206)(190 205)(191 204)(192 203)(193 202)(194 201)(195 200)

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

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

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

53 conjugacy classes

class 1 2A2B2C4A4B4C4D4E8A8B13A···13F26A···26F52A···52F52G···52R104A···104L
order1222444448813···1326···2652···5252···52104···104
size11265224426524522···22···24···48···84···4

53 irreducible representations

dim1111111122222444
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D13D26D26C8.C22D4×D13Q16⋊D13
kernelQ16⋊D13C8⋊D13C104⋊C2Q8⋊D13C13⋊Q16C13×Q16Q8×D13D52⋊C2Dic13D26Q16C8Q8C13C2C1
# reps111111111166121612

Matrix representation of Q16⋊D13 in GL4(𝔽313) generated by

008645
008183
255220225117
8313619688
,
21316824484
10910026301
1404114145
199179168299
,
115100
20014600
0001
00312261
,
14631200
3116700
0001
0010
G:=sub<GL(4,GF(313))| [0,0,255,83,0,0,220,136,86,81,225,196,45,83,117,88],[213,109,140,199,168,100,41,179,244,26,14,168,84,301,145,299],[115,200,0,0,1,146,0,0,0,0,0,312,0,0,1,261],[146,31,0,0,312,167,0,0,0,0,0,1,0,0,1,0] >;

Q16⋊D13 in GAP, Magma, Sage, TeX

Q_{16}\rtimes D_{13}
% in TeX

G:=Group("Q16:D13");
// GroupNames label

G:=SmallGroup(416,139);
// by ID

G=gap.SmallGroup(416,139);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,362,116,86,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^13=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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