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G = D4×C59order 472 = 23·59

Direct product of C59 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C59, C4⋊C118, C2363C2, C22⋊C118, C118.6C22, (C2×C118)⋊1C2, C2.1(C2×C118), SmallGroup(472,9)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C59
C1C2C118C2×C118 — D4×C59
C1C2 — D4×C59
C1C118 — D4×C59

Generators and relations for D4×C59
 G = < a,b,c | a59=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C118
2C118

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

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

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

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

295 conjugacy classes

class 1 2A2B2C 4 59A···59BF118A···118BF118BG···118FR236A···236BF
order1222459···59118···118118···118236···236
size112221···11···12···22···2

295 irreducible representations

dim11111122
type++++
imageC1C2C2C59C118C118D4D4×C59
kernelD4×C59C236C2×C118D4C4C22C59C1
# reps1125858116158

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

4790
0479
,
623508
70786
,
1623
0708
G:=sub<GL(2,GF(709))| [479,0,0,479],[623,707,508,86],[1,0,623,708] >;

D4×C59 in GAP, Magma, Sage, TeX

D_4\times C_{59}
% in TeX

G:=Group("D4xC59");
// GroupNames label

G:=SmallGroup(472,9);
// by ID

G=gap.SmallGroup(472,9);
# by ID

G:=PCGroup([4,-2,-2,-59,-2,1905]);
// Polycyclic

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

Export

Subgroup lattice of D4×C59 in TeX

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