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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22⋊C4×C26
 Chief series C1 — C2 — C22 — C2×C26 — C2×C52 — C13×C22⋊C4 — C22⋊C4×C26
 Lower central C1 — C2 — C22⋊C4×C26
 Upper central C1 — C22×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, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C13, C22⋊C4, C22×C4, C24, C26, C26, C26, C2×C22⋊C4, C52, C2×C26, C2×C26, C2×C26, C2×C52, C2×C52, C22×C26, C22×C26, C22×C26, C13×C22⋊C4, C22×C52, C23×C26, C22⋊C4×C26
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C13, C22⋊C4, C22×C4, C2×D4, C26, C2×C22⋊C4, C52, C2×C26, C2×C52, D4×C13, C22×C26, C13×C22⋊C4, C22×C52, D4×C26, 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 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(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)(105 202)(106 203)(107 204)(108 205)(109 206)(110 207)(111 208)(112 183)(113 184)(114 185)(115 186)(116 187)(117 188)(118 189)(119 190)(120 191)(121 192)(122 193)(123 194)(124 195)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(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)(157 170)(158 171)(159 172)(160 173)(161 174)(162 175)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(169 182)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 176)(28 177)(29 178)(30 179)(31 180)(32 181)(33 182)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(43 166)(44 167)(45 168)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(105 189)(106 190)(107 191)(108 192)(109 193)(110 194)(111 195)(112 196)(113 197)(114 198)(115 199)(116 200)(117 201)(118 202)(119 203)(120 204)(121 205)(122 206)(123 207)(124 208)(125 183)(126 184)(127 185)(128 186)(129 187)(130 188)
(1 76 199 44)(2 77 200 45)(3 78 201 46)(4 53 202 47)(5 54 203 48)(6 55 204 49)(7 56 205 50)(8 57 206 51)(9 58 207 52)(10 59 208 27)(11 60 183 28)(12 61 184 29)(13 62 185 30)(14 63 186 31)(15 64 187 32)(16 65 188 33)(17 66 189 34)(18 67 190 35)(19 68 191 36)(20 69 192 37)(21 70 193 38)(22 71 194 39)(23 72 195 40)(24 73 196 41)(25 74 197 42)(26 75 198 43)(79 139 115 167)(80 140 116 168)(81 141 117 169)(82 142 118 170)(83 143 119 171)(84 144 120 172)(85 145 121 173)(86 146 122 174)(87 147 123 175)(88 148 124 176)(89 149 125 177)(90 150 126 178)(91 151 127 179)(92 152 128 180)(93 153 129 181)(94 154 130 182)(95 155 105 157)(96 156 106 158)(97 131 107 159)(98 132 108 160)(99 133 109 161)(100 134 110 162)(101 135 111 163)(102 136 112 164)(103 137 113 165)(104 138 114 166)

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

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

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

260 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 13A ··· 13L 26A ··· 26CF 26CG ··· 26EB 52A ··· 52CR order 1 2 ··· 2 2 2 2 2 4 ··· 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C4 C13 C26 C26 C26 C52 D4 D4×C13 kernel C22⋊C4×C26 C13×C22⋊C4 C22×C52 C23×C26 C22×C26 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C23 C2×C26 C22 # reps 1 4 2 1 8 12 48 24 12 96 4 48

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

 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 0 0 0 4 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 2 0 0 19 49
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

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