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G = C22⋊C4×C26order 416 = 25·13

Direct product of C26 and C22⋊C4

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

Aliases: C22⋊C4×C26, C24.C26, C232C52, C2.1(D4×C26), (C22×C52)⋊3C2, C222(C2×C52), (C22×C26)⋊6C4, (C22×C4)⋊1C26, C26.64(C2×D4), (C2×C26).50D4, (C2×C52)⋊11C22, (C23×C26).1C2, C2.1(C22×C52), C23.5(C2×C26), (C2×C26).70C23, C26.42(C22×C4), C22.12(D4×C13), C22.4(C22×C26), (C22×C26).24C22, (C2×C4)⋊3(C2×C26), (C2×C26)⋊10(C2×C4), SmallGroup(416,176)

Series: Derived Chief Lower central Upper central

C1C2 — C22⋊C4×C26
C1C2C22C2×C26C2×C52C13×C22⋊C4 — C22⋊C4×C26
C1C2 — C22⋊C4×C26
C1C22×C26 — C22⋊C4×C26

Generators and relations for C22⋊C4×C26
 G = < a,b,c,d | a26=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 188 in 132 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C22, C22 [×10], C22 [×12], C2×C4 [×4], C2×C4 [×4], C23, C23 [×6], C23 [×4], C13, C22⋊C4 [×4], C22×C4 [×2], C24, C26, C26 [×6], C26 [×4], C2×C22⋊C4, C52 [×4], C2×C26, C2×C26 [×10], C2×C26 [×12], C2×C52 [×4], C2×C52 [×4], C22×C26, C22×C26 [×6], C22×C26 [×4], C13×C22⋊C4 [×4], C22×C52 [×2], C23×C26, C22⋊C4×C26
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C13, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C26 [×7], C2×C22⋊C4, C52 [×4], C2×C26 [×7], C2×C52 [×6], D4×C13 [×4], C22×C26, C13×C22⋊C4 [×4], C22×C52, D4×C26 [×2], C22⋊C4×C26

Smallest permutation representation of C22⋊C4×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)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 127)(15 128)(16 129)(17 130)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 113)(27 172)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 180)(36 181)(37 182)(38 157)(39 158)(40 159)(41 160)(42 161)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 169)(51 170)(52 171)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)(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)(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)
(1 127)(2 128)(3 129)(4 130)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 159)(28 160)(29 161)(30 162)(31 163)(32 164)(33 165)(34 166)(35 167)(36 168)(37 169)(38 170)(39 171)(40 172)(41 173)(42 174)(43 175)(44 176)(45 177)(46 178)(47 179)(48 180)(49 181)(50 182)(51 157)(52 158)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(131 206)(132 207)(133 208)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 190)(142 191)(143 192)(144 193)(145 194)(146 195)(147 196)(148 197)(149 198)(150 199)(151 200)(152 201)(153 202)(154 203)(155 204)(156 205)
(1 196 159 84)(2 197 160 85)(3 198 161 86)(4 199 162 87)(5 200 163 88)(6 201 164 89)(7 202 165 90)(8 203 166 91)(9 204 167 92)(10 205 168 93)(11 206 169 94)(12 207 170 95)(13 208 171 96)(14 183 172 97)(15 184 173 98)(16 185 174 99)(17 186 175 100)(18 187 176 101)(19 188 177 102)(20 189 178 103)(21 190 179 104)(22 191 180 79)(23 192 181 80)(24 193 182 81)(25 194 157 82)(26 195 158 83)(27 73 127 147)(28 74 128 148)(29 75 129 149)(30 76 130 150)(31 77 105 151)(32 78 106 152)(33 53 107 153)(34 54 108 154)(35 55 109 155)(36 56 110 156)(37 57 111 131)(38 58 112 132)(39 59 113 133)(40 60 114 134)(41 61 115 135)(42 62 116 136)(43 63 117 137)(44 64 118 138)(45 65 119 139)(46 66 120 140)(47 67 121 141)(48 68 122 142)(49 69 123 143)(50 70 124 144)(51 71 125 145)(52 72 126 146)

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

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

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

260 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H13A···13L26A···26CF26CG···26EB52A···52CR
order12···222224···413···1326···2626···2652···52
size11···122222···21···11···12···22···2

260 irreducible representations

dim111111111122
type+++++
imageC1C2C2C2C4C13C26C26C26C52D4D4×C13
kernelC22⋊C4×C26C13×C22⋊C4C22×C52C23×C26C22×C26C2×C22⋊C4C22⋊C4C22×C4C24C23C2×C26C22
# reps142181248241296448

Matrix representation of C22⋊C4×C26 in GL4(𝔽53) generated by

52000
05200
00150
00015
,
1000
0100
00520
0041
,
1000
0100
00520
00052
,
52000
02300
0042
001949
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,15,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,52,4,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,23,0,0,0,0,4,19,0,0,2,49] >;

C22⋊C4×C26 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_{26}
% in TeX

G:=Group("C2^2:C4xC26");
// GroupNames label

G:=SmallGroup(416,176);
// by ID

G=gap.SmallGroup(416,176);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273]);
// Polycyclic

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

׿
×
𝔽