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G = D26.6D4order 416 = 25·13

2nd non-split extension by D26 of D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D26.6D4, D4.5D26, C8.11D26, Q8.2D26, SD163D13, C52.7C23, D52.3C22, C104.11C22, Dic13.25D4, Dic26.3C22, D4⋊D134C2, (C8×D13)⋊5C2, C133(C4○D8), C104⋊C26C2, C13⋊Q162C2, C26.33(C2×D4), C2.21(D4×D13), D42D133C2, D52⋊C22C2, (C13×SD16)⋊4C2, C4.7(C22×D13), C132C8.6C22, (D4×C13).5C22, (Q8×C13).2C22, (C4×D13).18C22, SmallGroup(416,137)

Series: Derived Chief Lower central Upper central

C1C52 — D26.6D4
C1C13C26C52C4×D13D42D13 — D26.6D4
C13C26C52 — D26.6D4
C1C2C4SD16

Generators and relations for D26.6D4
 G = < a,b,c,d | a26=b2=d2=1, c4=a13, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a13b, dcd=c3 >

Subgroups: 464 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C8, C8, C2×C4 [×3], D4, D4 [×3], Q8, Q8, C13, C2×C8, D8, SD16, SD16, Q16, C4○D4 [×2], D13 [×2], C26, C26, C4○D8, Dic13, Dic13, C52, C52, D26, D26, C2×C26, C132C8, C104, Dic26, C4×D13, C4×D13, D52, D52, C2×Dic13, C13⋊D4, D4×C13, Q8×C13, C8×D13, C104⋊C2, D4⋊D13, C13⋊Q16, C13×SD16, D42D13, D52⋊C2, D26.6D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C4○D8, D26 [×3], C22×D13, D4×D13, D26.6D4

Smallest permutation representation of D26.6D4
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 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(51 52)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(72 78)(73 77)(74 76)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(96 104)(97 103)(98 102)(99 101)(105 127)(106 126)(107 125)(108 124)(109 123)(110 122)(111 121)(112 120)(113 119)(114 118)(115 117)(128 130)(131 153)(132 152)(133 151)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(141 143)(154 156)(157 162)(158 161)(159 160)(163 182)(164 181)(165 180)(166 179)(167 178)(168 177)(169 176)(170 175)(171 174)(172 173)(183 202)(184 201)(185 200)(186 199)(187 198)(188 197)(189 196)(190 195)(191 194)(192 193)(203 208)(204 207)(205 206)
(1 206 52 173 14 193 39 160)(2 207 27 174 15 194 40 161)(3 208 28 175 16 195 41 162)(4 183 29 176 17 196 42 163)(5 184 30 177 18 197 43 164)(6 185 31 178 19 198 44 165)(7 186 32 179 20 199 45 166)(8 187 33 180 21 200 46 167)(9 188 34 181 22 201 47 168)(10 189 35 182 23 202 48 169)(11 190 36 157 24 203 49 170)(12 191 37 158 25 204 50 171)(13 192 38 159 26 205 51 172)(53 133 91 120 66 146 104 107)(54 134 92 121 67 147 79 108)(55 135 93 122 68 148 80 109)(56 136 94 123 69 149 81 110)(57 137 95 124 70 150 82 111)(58 138 96 125 71 151 83 112)(59 139 97 126 72 152 84 113)(60 140 98 127 73 153 85 114)(61 141 99 128 74 154 86 115)(62 142 100 129 75 155 87 116)(63 143 101 130 76 156 88 117)(64 144 102 105 77 131 89 118)(65 145 103 106 78 132 90 119)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 121)(26 122)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 167)(64 168)(65 169)(66 170)(67 171)(68 172)(69 173)(70 174)(71 175)(72 176)(73 177)(74 178)(75 179)(76 180)(77 181)(78 182)(79 204)(80 205)(81 206)(82 207)(83 208)(84 183)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 191)(93 192)(94 193)(95 194)(96 195)(97 196)(98 197)(99 198)(100 199)(101 200)(102 201)(103 202)(104 203)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D13A···13F26A···26F26G···26L52A···52F52G···52L104A···104L
order1222244444888813···1326···2626···2652···5252···52104···104
size1142652241313522226262···22···28···84···48···84···4

56 irreducible representations

dim11111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D13C4○D8D26D26D26D4×D13D26.6D4
kernelD26.6D4C8×D13C104⋊C2D4⋊D13C13⋊Q16C13×SD16D42D13D52⋊C2Dic13D26SD16C13C8D4Q8C2C1
# reps111111111164666612

Matrix representation of D26.6D4 in GL4(𝔽313) generated by

30030000
133700
003120
000312
,
30030000
371300
003120
002801
,
312000
031200
001250
001025
,
312000
031200
00201187
00107112
G:=sub<GL(4,GF(313))| [300,13,0,0,300,37,0,0,0,0,312,0,0,0,0,312],[300,37,0,0,300,13,0,0,0,0,312,280,0,0,0,1],[312,0,0,0,0,312,0,0,0,0,125,102,0,0,0,5],[312,0,0,0,0,312,0,0,0,0,201,107,0,0,187,112] >;

D26.6D4 in GAP, Magma, Sage, TeX

D_{26}._6D_4
% in TeX

G:=Group("D26.6D4");
// GroupNames label

G:=SmallGroup(416,137);
// by ID

G=gap.SmallGroup(416,137);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,362,116,86,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=d^2=1,c^4=a^13,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^13*b,d*c*d=c^3>;
// generators/relations

׿
×
𝔽