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G = C4○D4×C26order 416 = 25·13

Direct product of C26 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C4○D4×C26, C52.55C23, C26.18C24, (C2×D4)⋊7C26, D43(C2×C26), (C2×Q8)⋊6C26, Q83(C2×C26), (D4×C26)⋊16C2, (C22×C4)⋊6C26, (Q8×C26)⋊13C2, (C2×C52)⋊16C22, (C22×C52)⋊13C2, (C2×C26).6C23, C4.8(C22×C26), C2.3(C23×C26), (D4×C13)⋊12C22, C23.11(C2×C26), (Q8×C13)⋊11C22, C22.1(C22×C26), (C22×C26).30C22, (C2×C4)⋊5(C2×C26), SmallGroup(416,230)

Series: Derived Chief Lower central Upper central

C1C2 — C4○D4×C26
C1C2C26C2×C26D4×C13C13×C4○D4 — C4○D4×C26
C1C2 — C4○D4×C26
C1C2×C52 — C4○D4×C26

Generators and relations for C4○D4×C26
 G = < a,b,c,d | a26=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C22, C22 [×6], C22 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C13, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C26, C26 [×2], C26 [×6], C2×C4○D4, C52 [×8], C2×C26, C2×C26 [×6], C2×C26 [×6], C2×C52, C2×C52 [×15], D4×C13 [×12], Q8×C13 [×4], C22×C26 [×3], C22×C52 [×3], D4×C26 [×3], Q8×C26, C13×C4○D4 [×8], C4○D4×C26
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C13, C4○D4 [×2], C24, C26 [×15], C2×C4○D4, C2×C26 [×35], C22×C26 [×15], C13×C4○D4 [×2], C23×C26, C4○D4×C26

Smallest permutation representation of C4○D4×C26
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 34 68 175)(2 35 69 176)(3 36 70 177)(4 37 71 178)(5 38 72 179)(6 39 73 180)(7 40 74 181)(8 41 75 182)(9 42 76 157)(10 43 77 158)(11 44 78 159)(12 45 53 160)(13 46 54 161)(14 47 55 162)(15 48 56 163)(16 49 57 164)(17 50 58 165)(18 51 59 166)(19 52 60 167)(20 27 61 168)(21 28 62 169)(22 29 63 170)(23 30 64 171)(24 31 65 172)(25 32 66 173)(26 33 67 174)(79 204 135 125)(80 205 136 126)(81 206 137 127)(82 207 138 128)(83 208 139 129)(84 183 140 130)(85 184 141 105)(86 185 142 106)(87 186 143 107)(88 187 144 108)(89 188 145 109)(90 189 146 110)(91 190 147 111)(92 191 148 112)(93 192 149 113)(94 193 150 114)(95 194 151 115)(96 195 152 116)(97 196 153 117)(98 197 154 118)(99 198 155 119)(100 199 156 120)(101 200 131 121)(102 201 132 122)(103 202 133 123)(104 203 134 124)
(1 194 68 115)(2 195 69 116)(3 196 70 117)(4 197 71 118)(5 198 72 119)(6 199 73 120)(7 200 74 121)(8 201 75 122)(9 202 76 123)(10 203 77 124)(11 204 78 125)(12 205 53 126)(13 206 54 127)(14 207 55 128)(15 208 56 129)(16 183 57 130)(17 184 58 105)(18 185 59 106)(19 186 60 107)(20 187 61 108)(21 188 62 109)(22 189 63 110)(23 190 64 111)(24 191 65 112)(25 192 66 113)(26 193 67 114)(27 144 168 88)(28 145 169 89)(29 146 170 90)(30 147 171 91)(31 148 172 92)(32 149 173 93)(33 150 174 94)(34 151 175 95)(35 152 176 96)(36 153 177 97)(37 154 178 98)(38 155 179 99)(39 156 180 100)(40 131 181 101)(41 132 182 102)(42 133 157 103)(43 134 158 104)(44 135 159 79)(45 136 160 80)(46 137 161 81)(47 138 162 82)(48 139 163 83)(49 140 164 84)(50 141 165 85)(51 142 166 86)(52 143 167 87)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 53)(26 54)(27 181)(28 182)(29 157)(30 158)(31 159)(32 160)(33 161)(34 162)(35 163)(36 164)(37 165)(38 166)(39 167)(40 168)(41 169)(42 170)(43 171)(44 172)(45 173)(46 174)(47 175)(48 176)(49 177)(50 178)(51 179)(52 180)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(105 118)(106 119)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(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)(183 196)(184 197)(185 198)(186 199)(187 200)(188 201)(189 202)(190 203)(191 204)(192 205)(193 206)(194 207)(195 208)

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

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

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

260 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J13A···13L26A···26AJ26AK···26DD52A···52AV52AW···52DP
order12222···244444···413···1326···2626···2652···5252···52
size11112···211112···21···11···12···21···12···2

260 irreducible representations

dim111111111122
type+++++
imageC1C2C2C2C2C13C26C26C26C26C4○D4C13×C4○D4
kernelC4○D4×C26C22×C52D4×C26Q8×C26C13×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C26C2
# reps133181236361296448

Matrix representation of C4○D4×C26 in GL3(𝔽53) generated by

5200
0460
0046
,
100
0300
0030
,
100
0362
01417
,
100
0520
0361
G:=sub<GL(3,GF(53))| [52,0,0,0,46,0,0,0,46],[1,0,0,0,30,0,0,0,30],[1,0,0,0,36,14,0,2,17],[1,0,0,0,52,36,0,0,1] >;

C4○D4×C26 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{26}
% in TeX

G:=Group("C4oD4xC26");
// GroupNames label

G:=SmallGroup(416,230);
// by ID

G=gap.SmallGroup(416,230);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-13,-2,2521,950]);
// Polycyclic

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

׿
×
𝔽