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G = C4⋊C47D13order 416 = 25·13

1st semidirect product of C4⋊C4 and D13 acting through Inn(C4⋊C4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C47D13, (C4×D13)⋊2C4, C52.32(C2×C4), C523C412C2, D26.9(C2×C4), (C2×C4).44D26, C4.14(C4×D13), C134(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(D42D13), 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

C1C26 — C4⋊C47D13
C1C13C26C2×C26C22×D13C2×C4×D13 — C4⋊C47D13
C13C26 — C4⋊C47D13
C1C22C4⋊C4

Generators and relations for C4⋊C47D13
 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 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C13, C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C22×C4, D13 [×2], C26 [×3], C42⋊C2, Dic13 [×2], Dic13 [×2], C52 [×2], C52 [×2], D26 [×2], D26 [×2], C2×C26, C4×D13 [×4], C2×Dic13, C2×Dic13 [×2], C2×C52, C2×C52 [×2], C22×D13, C4×Dic13 [×2], C523C4, D26⋊C4 [×2], C13×C4⋊C4, C2×C4×D13, C4⋊C47D13
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], D13, C42⋊C2, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D42D13, D52⋊C2, C4⋊C47D13

Smallest permutation representation of C4⋊C47D13
On 208 points
Generators in S208
(1 156 23 131)(2 144 24 132)(3 145 25 133)(4 146 26 134)(5 147 14 135)(6 148 15 136)(7 149 16 137)(8 150 17 138)(9 151 18 139)(10 152 19 140)(11 153 20 141)(12 154 21 142)(13 155 22 143)(27 128 52 112)(28 129 40 113)(29 130 41 114)(30 118 42 115)(31 119 43 116)(32 120 44 117)(33 121 45 105)(34 122 46 106)(35 123 47 107)(36 124 48 108)(37 125 49 109)(38 126 50 110)(39 127 51 111)(53 203 75 192)(54 204 76 193)(55 205 77 194)(56 206 78 195)(57 207 66 183)(58 208 67 184)(59 196 68 185)(60 197 69 186)(61 198 70 187)(62 199 71 188)(63 200 72 189)(64 201 73 190)(65 202 74 191)(79 174 95 165)(80 175 96 166)(81 176 97 167)(82 177 98 168)(83 178 99 169)(84 179 100 157)(85 180 101 158)(86 181 102 159)(87 182 103 160)(88 170 104 161)(89 171 92 162)(90 172 93 163)(91 173 94 164)
(1 79 33 54)(2 80 34 55)(3 81 35 56)(4 82 36 57)(5 83 37 58)(6 84 38 59)(7 85 39 60)(8 86 27 61)(9 87 28 62)(10 88 29 63)(11 89 30 64)(12 90 31 65)(13 91 32 53)(14 99 49 67)(15 100 50 68)(16 101 51 69)(17 102 52 70)(18 103 40 71)(19 104 41 72)(20 92 42 73)(21 93 43 74)(22 94 44 75)(23 95 45 76)(24 96 46 77)(25 97 47 78)(26 98 48 66)(105 204 131 174)(106 205 132 175)(107 206 133 176)(108 207 134 177)(109 208 135 178)(110 196 136 179)(111 197 137 180)(112 198 138 181)(113 199 139 182)(114 200 140 170)(115 201 141 171)(116 202 142 172)(117 203 143 173)(118 190 153 162)(119 191 154 163)(120 192 155 164)(121 193 156 165)(122 194 144 166)(123 195 145 167)(124 183 146 168)(125 184 147 169)(126 185 148 157)(127 186 149 158)(128 187 150 159)(129 188 151 160)(130 189 152 161)
(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 18)(15 17)(19 26)(20 25)(21 24)(22 23)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(40 49)(41 48)(42 47)(43 46)(44 45)(50 52)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 78)(65 77)(79 94)(80 93)(81 92)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(118 123)(119 122)(120 121)(124 130)(125 129)(126 128)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(144 154)(145 153)(146 152)(147 151)(148 150)(155 156)(157 181)(158 180)(159 179)(160 178)(161 177)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)(169 182)(183 200)(184 199)(185 198)(186 197)(187 196)(188 208)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J4K4L4M4N13A···13F26A···26R52A···52AJ
order1222224···44444444413···1326···2652···52
size111126262···213131313262626262···22···24···4

80 irreducible representations

dim1111111222244
type++++++++-+
imageC1C2C2C2C2C2C4C4○D4D13D26C4×D13D42D13D52⋊C2
kernelC4⋊C47D13C4×Dic13C523C4D26⋊C4C13×C4⋊C4C2×C4×D13C4×D13C26C4⋊C4C2×C4C4C2C2
# reps121211846182466

Matrix representation of C4⋊C47D13 in GL4(𝔽53) generated by

1000
0100
00300
003223
,
23000
02300
00118
004752
,
37100
442700
0010
0001
,
183100
463500
0010
004752
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⋊C47D13 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

׿
×
𝔽