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G = C59⋊D4order 472 = 23·59

The semidirect product of C59 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C592D4, C22⋊D59, Dic59⋊C2, D1182C2, C2.5D118, C118.5C22, (C2×C118)⋊2C2, SmallGroup(472,7)

Series: Derived Chief Lower central Upper central

C1C118 — C59⋊D4
C1C59C118D118 — C59⋊D4
C59C118 — C59⋊D4
C1C2C22

Generators and relations for C59⋊D4
 G = < a,b,c | a59=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
118C2
59C4
59C22
2D59
2C118
59D4

Smallest permutation representation of C59⋊D4
On 236 points
Generators in S236
(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)(178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236)
(1 126 69 227)(2 125 70 226)(3 124 71 225)(4 123 72 224)(5 122 73 223)(6 121 74 222)(7 120 75 221)(8 119 76 220)(9 177 77 219)(10 176 78 218)(11 175 79 217)(12 174 80 216)(13 173 81 215)(14 172 82 214)(15 171 83 213)(16 170 84 212)(17 169 85 211)(18 168 86 210)(19 167 87 209)(20 166 88 208)(21 165 89 207)(22 164 90 206)(23 163 91 205)(24 162 92 204)(25 161 93 203)(26 160 94 202)(27 159 95 201)(28 158 96 200)(29 157 97 199)(30 156 98 198)(31 155 99 197)(32 154 100 196)(33 153 101 195)(34 152 102 194)(35 151 103 193)(36 150 104 192)(37 149 105 191)(38 148 106 190)(39 147 107 189)(40 146 108 188)(41 145 109 187)(42 144 110 186)(43 143 111 185)(44 142 112 184)(45 141 113 183)(46 140 114 182)(47 139 115 181)(48 138 116 180)(49 137 117 179)(50 136 118 178)(51 135 60 236)(52 134 61 235)(53 133 62 234)(54 132 63 233)(55 131 64 232)(56 130 65 231)(57 129 66 230)(58 128 67 229)(59 127 68 228)
(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(79 118)(80 117)(81 116)(82 115)(83 114)(84 113)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(119 234)(120 233)(121 232)(122 231)(123 230)(124 229)(125 228)(126 227)(127 226)(128 225)(129 224)(130 223)(131 222)(132 221)(133 220)(134 219)(135 218)(136 217)(137 216)(138 215)(139 214)(140 213)(141 212)(142 211)(143 210)(144 209)(145 208)(146 207)(147 206)(148 205)(149 204)(150 203)(151 202)(152 201)(153 200)(154 199)(155 198)(156 197)(157 196)(158 195)(159 194)(160 193)(161 192)(162 191)(163 190)(164 189)(165 188)(166 187)(167 186)(168 185)(169 184)(170 183)(171 182)(172 181)(173 180)(174 179)(175 178)(176 236)(177 235)

G:=sub<Sym(236)| (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)(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236), (1,126,69,227)(2,125,70,226)(3,124,71,225)(4,123,72,224)(5,122,73,223)(6,121,74,222)(7,120,75,221)(8,119,76,220)(9,177,77,219)(10,176,78,218)(11,175,79,217)(12,174,80,216)(13,173,81,215)(14,172,82,214)(15,171,83,213)(16,170,84,212)(17,169,85,211)(18,168,86,210)(19,167,87,209)(20,166,88,208)(21,165,89,207)(22,164,90,206)(23,163,91,205)(24,162,92,204)(25,161,93,203)(26,160,94,202)(27,159,95,201)(28,158,96,200)(29,157,97,199)(30,156,98,198)(31,155,99,197)(32,154,100,196)(33,153,101,195)(34,152,102,194)(35,151,103,193)(36,150,104,192)(37,149,105,191)(38,148,106,190)(39,147,107,189)(40,146,108,188)(41,145,109,187)(42,144,110,186)(43,143,111,185)(44,142,112,184)(45,141,113,183)(46,140,114,182)(47,139,115,181)(48,138,116,180)(49,137,117,179)(50,136,118,178)(51,135,60,236)(52,134,61,235)(53,133,62,234)(54,132,63,233)(55,131,64,232)(56,130,65,231)(57,129,66,230)(58,128,67,229)(59,127,68,228), (2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(119,234)(120,233)(121,232)(122,231)(123,230)(124,229)(125,228)(126,227)(127,226)(128,225)(129,224)(130,223)(131,222)(132,221)(133,220)(134,219)(135,218)(136,217)(137,216)(138,215)(139,214)(140,213)(141,212)(142,211)(143,210)(144,209)(145,208)(146,207)(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)(163,190)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,236)(177,235)>;

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)(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236), (1,126,69,227)(2,125,70,226)(3,124,71,225)(4,123,72,224)(5,122,73,223)(6,121,74,222)(7,120,75,221)(8,119,76,220)(9,177,77,219)(10,176,78,218)(11,175,79,217)(12,174,80,216)(13,173,81,215)(14,172,82,214)(15,171,83,213)(16,170,84,212)(17,169,85,211)(18,168,86,210)(19,167,87,209)(20,166,88,208)(21,165,89,207)(22,164,90,206)(23,163,91,205)(24,162,92,204)(25,161,93,203)(26,160,94,202)(27,159,95,201)(28,158,96,200)(29,157,97,199)(30,156,98,198)(31,155,99,197)(32,154,100,196)(33,153,101,195)(34,152,102,194)(35,151,103,193)(36,150,104,192)(37,149,105,191)(38,148,106,190)(39,147,107,189)(40,146,108,188)(41,145,109,187)(42,144,110,186)(43,143,111,185)(44,142,112,184)(45,141,113,183)(46,140,114,182)(47,139,115,181)(48,138,116,180)(49,137,117,179)(50,136,118,178)(51,135,60,236)(52,134,61,235)(53,133,62,234)(54,132,63,233)(55,131,64,232)(56,130,65,231)(57,129,66,230)(58,128,67,229)(59,127,68,228), (2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(119,234)(120,233)(121,232)(122,231)(123,230)(124,229)(125,228)(126,227)(127,226)(128,225)(129,224)(130,223)(131,222)(132,221)(133,220)(134,219)(135,218)(136,217)(137,216)(138,215)(139,214)(140,213)(141,212)(142,211)(143,210)(144,209)(145,208)(146,207)(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)(163,190)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,236)(177,235) );

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),(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236)], [(1,126,69,227),(2,125,70,226),(3,124,71,225),(4,123,72,224),(5,122,73,223),(6,121,74,222),(7,120,75,221),(8,119,76,220),(9,177,77,219),(10,176,78,218),(11,175,79,217),(12,174,80,216),(13,173,81,215),(14,172,82,214),(15,171,83,213),(16,170,84,212),(17,169,85,211),(18,168,86,210),(19,167,87,209),(20,166,88,208),(21,165,89,207),(22,164,90,206),(23,163,91,205),(24,162,92,204),(25,161,93,203),(26,160,94,202),(27,159,95,201),(28,158,96,200),(29,157,97,199),(30,156,98,198),(31,155,99,197),(32,154,100,196),(33,153,101,195),(34,152,102,194),(35,151,103,193),(36,150,104,192),(37,149,105,191),(38,148,106,190),(39,147,107,189),(40,146,108,188),(41,145,109,187),(42,144,110,186),(43,143,111,185),(44,142,112,184),(45,141,113,183),(46,140,114,182),(47,139,115,181),(48,138,116,180),(49,137,117,179),(50,136,118,178),(51,135,60,236),(52,134,61,235),(53,133,62,234),(54,132,63,233),(55,131,64,232),(56,130,65,231),(57,129,66,230),(58,128,67,229),(59,127,68,228)], [(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(79,118),(80,117),(81,116),(82,115),(83,114),(84,113),(85,112),(86,111),(87,110),(88,109),(89,108),(90,107),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99),(119,234),(120,233),(121,232),(122,231),(123,230),(124,229),(125,228),(126,227),(127,226),(128,225),(129,224),(130,223),(131,222),(132,221),(133,220),(134,219),(135,218),(136,217),(137,216),(138,215),(139,214),(140,213),(141,212),(142,211),(143,210),(144,209),(145,208),(146,207),(147,206),(148,205),(149,204),(150,203),(151,202),(152,201),(153,200),(154,199),(155,198),(156,197),(157,196),(158,195),(159,194),(160,193),(161,192),(162,191),(163,190),(164,189),(165,188),(166,187),(167,186),(168,185),(169,184),(170,183),(171,182),(172,181),(173,180),(174,179),(175,178),(176,236),(177,235)])

121 conjugacy classes

class 1 2A2B2C 4 59A···59AC118A···118CI
order1222459···59118···118
size1121181182···22···2

121 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D59D118C59⋊D4
kernelC59⋊D4Dic59D118C2×C118C59C22C2C1
# reps11111292958

Matrix representation of C59⋊D4 in GL2(𝔽709) generated by

4901
135226
,
634416
50875
,
394142
180315
G:=sub<GL(2,GF(709))| [490,135,1,226],[634,508,416,75],[394,180,142,315] >;

C59⋊D4 in GAP, Magma, Sage, TeX

C_{59}\rtimes D_4
% in TeX

G:=Group("C59:D4");
// GroupNames label

G:=SmallGroup(472,7);
// by ID

G=gap.SmallGroup(472,7);
# by ID

G:=PCGroup([4,-2,-2,-2,-59,49,7427]);
// Polycyclic

G:=Group<a,b,c|a^59=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C59⋊D4 in TeX

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