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G = C4⋊C4×D13order 416 = 25·13

Direct product of C4⋊C4 and D13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4×D13, D26.5Q8, D26.22D4, C524(C2×C4), C43(C4×D13), (C4×D13)⋊1C4, C2.3(D4×D13), C2.2(Q8×D13), C523C411C2, C26.23(C2×D4), (C2×C4).30D26, C26.12(C2×Q8), Dic134(C2×C4), D26.19(C2×C4), C26.D411C2, (C2×C52).23C22, (C2×C26).32C23, C26.22(C22×C4), C22.16(C22×D13), (C2×Dic13).32C22, (C22×D13).43C22, C132(C2×C4⋊C4), (C13×C4⋊C4)⋊2C2, (C2×C4×D13).1C2, C2.11(C2×C4×D13), SmallGroup(416,112)

Series: Derived Chief Lower central Upper central

C1C26 — C4⋊C4×D13
C1C13C26C2×C26C22×D13C2×C4×D13 — C4⋊C4×D13
C13C26 — C4⋊C4×D13
C1C22C4⋊C4

Generators and relations for C4⋊C4×D13
 G = < a,b,c,d | a4=b4=c13=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 576 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, C13, C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], D13 [×4], C26 [×3], C2×C4⋊C4, Dic13 [×2], Dic13 [×2], C52 [×2], C52 [×2], D26 [×6], C2×C26, C4×D13 [×4], C4×D13 [×4], C2×Dic13, C2×Dic13 [×2], C2×C52, C2×C52 [×2], C22×D13, C26.D4 [×2], C523C4, C13×C4⋊C4, C2×C4×D13, C2×C4×D13 [×2], C4⋊C4×D13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D13, C2×C4⋊C4, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D4×D13, Q8×D13, C4⋊C4×D13

Smallest permutation representation of C4⋊C4×D13
On 208 points
Generators in S208
(1 152 22 138)(2 153 23 139)(3 154 24 140)(4 155 25 141)(5 156 26 142)(6 144 14 143)(7 145 15 131)(8 146 16 132)(9 147 17 133)(10 148 18 134)(11 149 19 135)(12 150 20 136)(13 151 21 137)(27 120 45 115)(28 121 46 116)(29 122 47 117)(30 123 48 105)(31 124 49 106)(32 125 50 107)(33 126 51 108)(34 127 52 109)(35 128 40 110)(36 129 41 111)(37 130 42 112)(38 118 43 113)(39 119 44 114)(53 201 66 183)(54 202 67 184)(55 203 68 185)(56 204 69 186)(57 205 70 187)(58 206 71 188)(59 207 72 189)(60 208 73 190)(61 196 74 191)(62 197 75 192)(63 198 76 193)(64 199 77 194)(65 200 78 195)(79 179 104 163)(80 180 92 164)(81 181 93 165)(82 182 94 166)(83 170 95 167)(84 171 96 168)(85 172 97 169)(86 173 98 157)(87 174 99 158)(88 175 100 159)(89 176 101 160)(90 177 102 161)(91 178 103 162)
(1 79 30 60)(2 80 31 61)(3 81 32 62)(4 82 33 63)(5 83 34 64)(6 84 35 65)(7 85 36 53)(8 86 37 54)(9 87 38 55)(10 88 39 56)(11 89 27 57)(12 90 28 58)(13 91 29 59)(14 96 40 78)(15 97 41 66)(16 98 42 67)(17 99 43 68)(18 100 44 69)(19 101 45 70)(20 102 46 71)(21 103 47 72)(22 104 48 73)(23 92 49 74)(24 93 50 75)(25 94 51 76)(26 95 52 77)(105 208 138 179)(106 196 139 180)(107 197 140 181)(108 198 141 182)(109 199 142 170)(110 200 143 171)(111 201 131 172)(112 202 132 173)(113 203 133 174)(114 204 134 175)(115 205 135 176)(116 206 136 177)(117 207 137 178)(118 185 147 158)(119 186 148 159)(120 187 149 160)(121 188 150 161)(122 189 151 162)(123 190 152 163)(124 191 153 164)(125 192 154 165)(126 193 155 166)(127 194 156 167)(128 195 144 168)(129 183 145 169)(130 184 146 157)
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 52)(10 51)(11 50)(12 49)(13 48)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 39)(26 38)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(105 151)(106 150)(107 149)(108 148)(109 147)(110 146)(111 145)(112 144)(113 156)(114 155)(115 154)(116 153)(117 152)(118 142)(119 141)(120 140)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(127 133)(128 132)(129 131)(130 143)(157 200)(158 199)(159 198)(160 197)(161 196)(162 208)(163 207)(164 206)(165 205)(166 204)(167 203)(168 202)(169 201)(170 185)(171 184)(172 183)(173 195)(174 194)(175 193)(176 192)(177 191)(178 190)(179 189)(180 188)(181 187)(182 186)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G···4L13A···13F26A···26R52A···52AJ
order122222224···44···413···1326···2652···52
size1111131313132···226···262···22···24···4

80 irreducible representations

dim1111112222244
type++++++-+++-
imageC1C2C2C2C2C4D4Q8D13D26C4×D13D4×D13Q8×D13
kernelC4⋊C4×D13C26.D4C523C4C13×C4⋊C4C2×C4×D13C4×D13D26D26C4⋊C4C2×C4C4C2C2
# reps121138226182466

Matrix representation of C4⋊C4×D13 in GL4(𝔽53) generated by

1000
0100
00379
004816
,
23000
02300
005217
0001
,
0100
522600
0010
0001
,
05200
52000
00520
00052
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,37,48,0,0,9,16],[23,0,0,0,0,23,0,0,0,0,52,0,0,0,17,1],[0,52,0,0,1,26,0,0,0,0,1,0,0,0,0,1],[0,52,0,0,52,0,0,0,0,0,52,0,0,0,0,52] >;

C4⋊C4×D13 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times D_{13}
% in TeX

G:=Group("C4:C4xD13");
// GroupNames label

G:=SmallGroup(416,112);
// by ID

G=gap.SmallGroup(416,112);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,188,50,13829]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽