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G = D26⋊C8order 416 = 25·13

2nd semidirect product of D26 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D262C8, C26.3M4(2), Dic13.20D4, C26.3(C2×C8), (C2×C52).3C4, C131(C22⋊C8), C2.4(D13⋊C8), C26.2(C22⋊C4), (C22×D13).6C4, C2.1(D13.D4), C2.3(C52.C4), (C2×Dic13).51C22, (C2×C13⋊C8)⋊1C2, (C2×C4×D13).8C2, (C2×C4).3(C13⋊C4), (C2×C26).6(C2×C4), C22.11(C2×C13⋊C4), SmallGroup(416,78)

Series: Derived Chief Lower central Upper central

C1C26 — D26⋊C8
C1C13C26Dic13C2×Dic13C2×C13⋊C8 — D26⋊C8
C13C26 — D26⋊C8
C1C22C2×C4

Generators and relations for D26⋊C8
 G = < a,b,c | a26=b2=c8=1, bab=a-1, cac-1=a5, cbc-1=a17b >

26C2
26C2
2C4
13C4
13C22
13C22
13C4
26C22
26C22
2D13
2D13
13C23
13C2×C4
26C8
26C2×C4
26C8
26C2×C4
2D26
2C52
2D26
13C2×C8
13C22×C4
13C2×C8
2C13⋊C8
2C4×D13
2C13⋊C8
2C4×D13
13C22⋊C8

Smallest permutation representation of D26⋊C8
On 208 points
Generators in S208
(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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)(13 14)(27 31)(28 30)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(53 54)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(101 104)(102 103)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(131 153)(132 152)(133 151)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(141 143)(154 156)(157 181)(158 180)(159 179)(160 178)(161 177)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)(183 192)(184 191)(185 190)(186 189)(187 188)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)
(1 49 188 136 67 111 103 163)(2 44 187 141 68 106 102 168)(3 39 186 146 69 127 101 173)(4 34 185 151 70 122 100 178)(5 29 184 156 71 117 99 157)(6 50 183 135 72 112 98 162)(7 45 208 140 73 107 97 167)(8 40 207 145 74 128 96 172)(9 35 206 150 75 123 95 177)(10 30 205 155 76 118 94 182)(11 51 204 134 77 113 93 161)(12 46 203 139 78 108 92 166)(13 41 202 144 53 129 91 171)(14 36 201 149 54 124 90 176)(15 31 200 154 55 119 89 181)(16 52 199 133 56 114 88 160)(17 47 198 138 57 109 87 165)(18 42 197 143 58 130 86 170)(19 37 196 148 59 125 85 175)(20 32 195 153 60 120 84 180)(21 27 194 132 61 115 83 159)(22 48 193 137 62 110 82 164)(23 43 192 142 63 105 81 169)(24 38 191 147 64 126 80 174)(25 33 190 152 65 121 79 179)(26 28 189 131 66 116 104 158)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F8A···8H13A13B13C26A···26I52A···52L
order1222224444448···813131326···2652···52
size11112626221313131326···264444···44···4

44 irreducible representations

dim1111112244444
type+++++++
imageC1C2C2C4C4C8D4M4(2)C13⋊C4C2×C13⋊C4D13⋊C8C52.C4D13.D4
kernelD26⋊C8C2×C13⋊C8C2×C4×D13C2×C52C22×D13D26Dic13C26C2×C4C22C2C2C2
# reps1212282233666

Matrix representation of D26⋊C8 in GL6(𝔽313)

31200000
03120000
0070200199100
002138484213
00131230231101
002432132430
,
31200000
010000
0024302929
00100312212212
00182212312311
00701017172
,
010000
100000
00268185193311
00133304242310
0029615220138
009819618634

G:=sub<GL(6,GF(313))| [312,0,0,0,0,0,0,312,0,0,0,0,0,0,70,213,131,243,0,0,200,84,230,213,0,0,199,84,231,243,0,0,100,213,101,0],[312,0,0,0,0,0,0,1,0,0,0,0,0,0,243,100,182,70,0,0,0,312,212,101,0,0,29,212,312,71,0,0,29,212,311,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,268,133,296,98,0,0,185,304,152,196,0,0,193,242,20,186,0,0,311,310,138,34] >;

D26⋊C8 in GAP, Magma, Sage, TeX

D_{26}\rtimes C_8
% in TeX

G:=Group("D26:C8");
// GroupNames label

G:=SmallGroup(416,78);
// by ID

G=gap.SmallGroup(416,78);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,86,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^26=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^17*b>;
// generators/relations

Export

Subgroup lattice of D26⋊C8 in TeX

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