metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊7D13, (C4×D13)⋊2C4, C52.32(C2×C4), C52⋊3C4⋊12C2, D26.9(C2×C4), (C2×C4).44D26, C4.14(C4×D13), C13⋊4(C42⋊C2), (C4×Dic13)⋊13C2, D26⋊C4.4C2, C26.26(C4○D4), C26.23(C22×C4), (C2×C26).33C23, (C2×C52).56C22, C2.1(D52⋊C2), C2.4(D4⋊2D13), Dic13.21(C2×C4), C22.17(C22×D13), (C2×Dic13).33C22, (C22×D13).22C22, (C13×C4⋊C4)⋊3C2, (C2×C4×D13).2C2, C2.12(C2×C4×D13), SmallGroup(416,113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊7D13
G = < a,b,c,d | a4=b4=c13=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 464 in 76 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, C23, C13, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, D13, C26, C42⋊C2, Dic13, Dic13, C52, C52, D26, D26, C2×C26, C4×D13, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C22×D13, C4×Dic13, C52⋊3C4, D26⋊C4, C13×C4⋊C4, C2×C4×D13, C4⋊C4⋊7D13
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, D13, C42⋊C2, D26, C4×D13, C22×D13, C2×C4×D13, D4⋊2D13, D52⋊C2, C4⋊C4⋊7D13
►On 208 points
Generators in S208
(1 151 19 136)(2 152 20 137)(3 153 21 138)(4 154 22 139)(5 155 23 140)(6 156 24 141)(7 144 25 142)(8 145 26 143)(9 146 14 131)(10 147 15 132)(11 148 16 133)(12 149 17 134)(13 150 18 135)(27 130 44 114)(28 118 45 115)(29 119 46 116)(30 120 47 117)(31 121 48 105)(32 122 49 106)(33 123 50 107)(34 124 51 108)(35 125 52 109)(36 126 40 110)(37 127 41 111)(38 128 42 112)(39 129 43 113)(53 207 68 193)(54 208 69 194)(55 196 70 195)(56 197 71 183)(57 198 72 184)(58 199 73 185)(59 200 74 186)(60 201 75 187)(61 202 76 188)(62 203 77 189)(63 204 78 190)(64 205 66 191)(65 206 67 192)(79 172 104 157)(80 173 92 158)(81 174 93 159)(82 175 94 160)(83 176 95 161)(84 177 96 162)(85 178 97 163)(86 179 98 164)(87 180 99 165)(88 181 100 166)(89 182 101 167)(90 170 102 168)(91 171 103 169)
(1 91 36 61)(2 79 37 62)(3 80 38 63)(4 81 39 64)(5 82 27 65)(6 83 28 53)(7 84 29 54)(8 85 30 55)(9 86 31 56)(10 87 32 57)(11 88 33 58)(12 89 34 59)(13 90 35 60)(14 98 48 71)(15 99 49 72)(16 100 50 73)(17 101 51 74)(18 102 52 75)(19 103 40 76)(20 104 41 77)(21 92 42 78)(22 93 43 66)(23 94 44 67)(24 95 45 68)(25 96 46 69)(26 97 47 70)(105 197 131 179)(106 198 132 180)(107 199 133 181)(108 200 134 182)(109 201 135 170)(110 202 136 171)(111 203 137 172)(112 204 138 173)(113 205 139 174)(114 206 140 175)(115 207 141 176)(116 208 142 177)(117 196 143 178)(118 193 156 161)(119 194 144 162)(120 195 145 163)(121 183 146 164)(122 184 147 165)(123 185 148 166)(124 186 149 167)(125 187 150 168)(126 188 151 169)(127 189 152 157)(128 190 153 158)(129 191 154 159)(130 192 155 160)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 23)(15 22)(16 21)(17 20)(18 19)(24 26)(27 31)(28 30)(32 39)(33 38)(34 37)(35 36)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 70)(54 69)(55 68)(56 67)(57 66)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(79 101)(80 100)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 104)(90 103)(91 102)(105 114)(106 113)(107 112)(108 111)(109 110)(115 117)(118 120)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 143)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(183 206)(184 205)(185 204)(186 203)(187 202)(188 201)(189 200)(190 199)(191 198)(192 197)(193 196)(194 208)(195 207)
G:=sub<Sym(208)| (1,151,19,136)(2,152,20,137)(3,153,21,138)(4,154,22,139)(5,155,23,140)(6,156,24,141)(7,144,25,142)(8,145,26,143)(9,146,14,131)(10,147,15,132)(11,148,16,133)(12,149,17,134)(13,150,18,135)(27,130,44,114)(28,118,45,115)(29,119,46,116)(30,120,47,117)(31,121,48,105)(32,122,49,106)(33,123,50,107)(34,124,51,108)(35,125,52,109)(36,126,40,110)(37,127,41,111)(38,128,42,112)(39,129,43,113)(53,207,68,193)(54,208,69,194)(55,196,70,195)(56,197,71,183)(57,198,72,184)(58,199,73,185)(59,200,74,186)(60,201,75,187)(61,202,76,188)(62,203,77,189)(63,204,78,190)(64,205,66,191)(65,206,67,192)(79,172,104,157)(80,173,92,158)(81,174,93,159)(82,175,94,160)(83,176,95,161)(84,177,96,162)(85,178,97,163)(86,179,98,164)(87,180,99,165)(88,181,100,166)(89,182,101,167)(90,170,102,168)(91,171,103,169), (1,91,36,61)(2,79,37,62)(3,80,38,63)(4,81,39,64)(5,82,27,65)(6,83,28,53)(7,84,29,54)(8,85,30,55)(9,86,31,56)(10,87,32,57)(11,88,33,58)(12,89,34,59)(13,90,35,60)(14,98,48,71)(15,99,49,72)(16,100,50,73)(17,101,51,74)(18,102,52,75)(19,103,40,76)(20,104,41,77)(21,92,42,78)(22,93,43,66)(23,94,44,67)(24,95,45,68)(25,96,46,69)(26,97,47,70)(105,197,131,179)(106,198,132,180)(107,199,133,181)(108,200,134,182)(109,201,135,170)(110,202,136,171)(111,203,137,172)(112,204,138,173)(113,205,139,174)(114,206,140,175)(115,207,141,176)(116,208,142,177)(117,196,143,178)(118,193,156,161)(119,194,144,162)(120,195,145,163)(121,183,146,164)(122,184,147,165)(123,185,148,166)(124,186,149,167)(125,187,150,168)(126,188,151,169)(127,189,152,157)(128,190,153,158)(129,191,154,159)(130,192,155,160), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,23)(15,22)(16,21)(17,20)(18,19)(24,26)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,70)(54,69)(55,68)(56,67)(57,66)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,104)(90,103)(91,102)(105,114)(106,113)(107,112)(108,111)(109,110)(115,117)(118,120)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,143)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(183,206)(184,205)(185,204)(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)(193,196)(194,208)(195,207)>;
G:=Group( (1,151,19,136)(2,152,20,137)(3,153,21,138)(4,154,22,139)(5,155,23,140)(6,156,24,141)(7,144,25,142)(8,145,26,143)(9,146,14,131)(10,147,15,132)(11,148,16,133)(12,149,17,134)(13,150,18,135)(27,130,44,114)(28,118,45,115)(29,119,46,116)(30,120,47,117)(31,121,48,105)(32,122,49,106)(33,123,50,107)(34,124,51,108)(35,125,52,109)(36,126,40,110)(37,127,41,111)(38,128,42,112)(39,129,43,113)(53,207,68,193)(54,208,69,194)(55,196,70,195)(56,197,71,183)(57,198,72,184)(58,199,73,185)(59,200,74,186)(60,201,75,187)(61,202,76,188)(62,203,77,189)(63,204,78,190)(64,205,66,191)(65,206,67,192)(79,172,104,157)(80,173,92,158)(81,174,93,159)(82,175,94,160)(83,176,95,161)(84,177,96,162)(85,178,97,163)(86,179,98,164)(87,180,99,165)(88,181,100,166)(89,182,101,167)(90,170,102,168)(91,171,103,169), (1,91,36,61)(2,79,37,62)(3,80,38,63)(4,81,39,64)(5,82,27,65)(6,83,28,53)(7,84,29,54)(8,85,30,55)(9,86,31,56)(10,87,32,57)(11,88,33,58)(12,89,34,59)(13,90,35,60)(14,98,48,71)(15,99,49,72)(16,100,50,73)(17,101,51,74)(18,102,52,75)(19,103,40,76)(20,104,41,77)(21,92,42,78)(22,93,43,66)(23,94,44,67)(24,95,45,68)(25,96,46,69)(26,97,47,70)(105,197,131,179)(106,198,132,180)(107,199,133,181)(108,200,134,182)(109,201,135,170)(110,202,136,171)(111,203,137,172)(112,204,138,173)(113,205,139,174)(114,206,140,175)(115,207,141,176)(116,208,142,177)(117,196,143,178)(118,193,156,161)(119,194,144,162)(120,195,145,163)(121,183,146,164)(122,184,147,165)(123,185,148,166)(124,186,149,167)(125,187,150,168)(126,188,151,169)(127,189,152,157)(128,190,153,158)(129,191,154,159)(130,192,155,160), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,23)(15,22)(16,21)(17,20)(18,19)(24,26)(27,31)(28,30)(32,39)(33,38)(34,37)(35,36)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,70)(54,69)(55,68)(56,67)(57,66)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(79,101)(80,100)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,104)(90,103)(91,102)(105,114)(106,113)(107,112)(108,111)(109,110)(115,117)(118,120)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,143)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(183,206)(184,205)(185,204)(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)(193,196)(194,208)(195,207) );
G=PermutationGroup([[(1,151,19,136),(2,152,20,137),(3,153,21,138),(4,154,22,139),(5,155,23,140),(6,156,24,141),(7,144,25,142),(8,145,26,143),(9,146,14,131),(10,147,15,132),(11,148,16,133),(12,149,17,134),(13,150,18,135),(27,130,44,114),(28,118,45,115),(29,119,46,116),(30,120,47,117),(31,121,48,105),(32,122,49,106),(33,123,50,107),(34,124,51,108),(35,125,52,109),(36,126,40,110),(37,127,41,111),(38,128,42,112),(39,129,43,113),(53,207,68,193),(54,208,69,194),(55,196,70,195),(56,197,71,183),(57,198,72,184),(58,199,73,185),(59,200,74,186),(60,201,75,187),(61,202,76,188),(62,203,77,189),(63,204,78,190),(64,205,66,191),(65,206,67,192),(79,172,104,157),(80,173,92,158),(81,174,93,159),(82,175,94,160),(83,176,95,161),(84,177,96,162),(85,178,97,163),(86,179,98,164),(87,180,99,165),(88,181,100,166),(89,182,101,167),(90,170,102,168),(91,171,103,169)], [(1,91,36,61),(2,79,37,62),(3,80,38,63),(4,81,39,64),(5,82,27,65),(6,83,28,53),(7,84,29,54),(8,85,30,55),(9,86,31,56),(10,87,32,57),(11,88,33,58),(12,89,34,59),(13,90,35,60),(14,98,48,71),(15,99,49,72),(16,100,50,73),(17,101,51,74),(18,102,52,75),(19,103,40,76),(20,104,41,77),(21,92,42,78),(22,93,43,66),(23,94,44,67),(24,95,45,68),(25,96,46,69),(26,97,47,70),(105,197,131,179),(106,198,132,180),(107,199,133,181),(108,200,134,182),(109,201,135,170),(110,202,136,171),(111,203,137,172),(112,204,138,173),(113,205,139,174),(114,206,140,175),(115,207,141,176),(116,208,142,177),(117,196,143,178),(118,193,156,161),(119,194,144,162),(120,195,145,163),(121,183,146,164),(122,184,147,165),(123,185,148,166),(124,186,149,167),(125,187,150,168),(126,188,151,169),(127,189,152,157),(128,190,153,158),(129,191,154,159),(130,192,155,160)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,23),(15,22),(16,21),(17,20),(18,19),(24,26),(27,31),(28,30),(32,39),(33,38),(34,37),(35,36),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,70),(54,69),(55,68),(56,67),(57,66),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(79,101),(80,100),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,104),(90,103),(91,102),(105,114),(106,113),(107,112),(108,111),(109,110),(115,117),(118,120),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,143),(145,156),(146,155),(147,154),(148,153),(149,152),(150,151),(157,182),(158,181),(159,180),(160,179),(161,178),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,171),(169,170),(183,206),(184,205),(185,204),(186,203),(187,202),(188,201),(189,200),(190,199),(191,198),(192,197),(193,196),(194,208),(195,207)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 26 | 26 | 2 | ··· | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | D13 | D26 | C4×D13 | D4⋊2D13 | D52⋊C2 |
| kernel | C4⋊C4⋊7D13 | C4×Dic13 | C52⋊3C4 | D26⋊C4 | C13×C4⋊C4 | C2×C4×D13 | C4×D13 | C26 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 6 | 18 | 24 | 6 | 6 |
Matrix representation of C4⋊C4⋊7D13 ►in GL4(𝔽53) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 32 | 23 |
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 1 | 18 |
| 0 | 0 | 47 | 52 |
| 37 | 1 | 0 | 0 |
| 44 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 31 | 0 | 0 |
| 46 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 47 | 52 |
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,30,32,0,0,0,23],[23,0,0,0,0,23,0,0,0,0,1,47,0,0,18,52],[37,44,0,0,1,27,0,0,0,0,1,0,0,0,0,1],[18,46,0,0,31,35,0,0,0,0,1,47,0,0,0,52] >;
C4⋊C4⋊7D13 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_7D_{13} % in TeX
G:=Group("C4:C4:7D13"); // GroupNames label
G:=SmallGroup(416,113);
// by ID
G=gap.SmallGroup(416,113);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,362,188,50,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^13=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations