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G = C4×D57order 456 = 23·3·19

Direct product of C4 and D57

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D57, C762S3, C2282C2, C122D19, C6.9D38, C38.9D6, C2.1D114, Dic575C2, D114.2C2, C114.9C22, C192(C4×S3), C574(C2×C4), C32(C4×D19), SmallGroup(456,35)

Series: Derived Chief Lower central Upper central

C1C57 — C4×D57
C1C19C57C114D114 — C4×D57
C57 — C4×D57
C1C4

Generators and relations for C4×D57
 G = < a,b,c | a4=b57=c2=1, ab=ba, ac=ca, cbc=b-1 >

57C2
57C2
57C4
57C22
19S3
19S3
3D19
3D19
57C2×C4
19Dic3
19D6
3D38
3Dic19
19C4×S3
3C4×D19

Smallest permutation representation of C4×D57
On 228 points
Generators in S228
(1 188 102 152)(2 189 103 153)(3 190 104 154)(4 191 105 155)(5 192 106 156)(6 193 107 157)(7 194 108 158)(8 195 109 159)(9 196 110 160)(10 197 111 161)(11 198 112 162)(12 199 113 163)(13 200 114 164)(14 201 58 165)(15 202 59 166)(16 203 60 167)(17 204 61 168)(18 205 62 169)(19 206 63 170)(20 207 64 171)(21 208 65 115)(22 209 66 116)(23 210 67 117)(24 211 68 118)(25 212 69 119)(26 213 70 120)(27 214 71 121)(28 215 72 122)(29 216 73 123)(30 217 74 124)(31 218 75 125)(32 219 76 126)(33 220 77 127)(34 221 78 128)(35 222 79 129)(36 223 80 130)(37 224 81 131)(38 225 82 132)(39 226 83 133)(40 227 84 134)(41 228 85 135)(42 172 86 136)(43 173 87 137)(44 174 88 138)(45 175 89 139)(46 176 90 140)(47 177 91 141)(48 178 92 142)(49 179 93 143)(50 180 94 144)(51 181 95 145)(52 182 96 146)(53 183 97 147)(54 184 98 148)(55 185 99 149)(56 186 100 150)(57 187 101 151)
(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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 96)(7 95)(8 94)(9 93)(10 92)(11 91)(12 90)(13 89)(14 88)(15 87)(16 86)(17 85)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 114)(46 113)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 104)(56 103)(57 102)(115 224)(116 223)(117 222)(118 221)(119 220)(120 219)(121 218)(122 217)(123 216)(124 215)(125 214)(126 213)(127 212)(128 211)(129 210)(130 209)(131 208)(132 207)(133 206)(134 205)(135 204)(136 203)(137 202)(138 201)(139 200)(140 199)(141 198)(142 197)(143 196)(144 195)(145 194)(146 193)(147 192)(148 191)(149 190)(150 189)(151 188)(152 187)(153 186)(154 185)(155 184)(156 183)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 228)(169 227)(170 226)(171 225)

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B19A···19I38A···38I57A···57R76A···76R114A···114R228A···228AJ
order1222344446121219···1938···3857···5776···76114···114228···228
size11575721157572222···22···22···22···22···22···2

120 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3D6C4×S3D19D38D57C4×D19D114C4×D57
kernelC4×D57Dic57C228D114D57C76C38C19C12C6C4C3C2C1
# reps111141129918181836

Matrix representation of C4×D57 in GL3(𝔽229) generated by

12200
010
001
,
100
0140155
074149
,
100
0140155
0889
G:=sub<GL(3,GF(229))| [122,0,0,0,1,0,0,0,1],[1,0,0,0,140,74,0,155,149],[1,0,0,0,140,8,0,155,89] >;

C4×D57 in GAP, Magma, Sage, TeX

C_4\times D_{57}
% in TeX

G:=Group("C4xD57");
// GroupNames label

G:=SmallGroup(456,35);
// by ID

G=gap.SmallGroup(456,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-19,26,323,10804]);
// Polycyclic

G:=Group<a,b,c|a^4=b^57=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D57 in TeX

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