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G = C3×D76order 456 = 23·3·19

Direct product of C3 and D76

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

Aliases: C3×D76, C575D4, C765C6, C2283C2, D384C6, C123D19, C6.15D38, C114.15C22, C4⋊(C3×D19), C194(C3×D4), (C6×D19)⋊4C2, C2.4(C6×D19), C38.11(C2×C6), SmallGroup(456,26)

Series: Derived Chief Lower central Upper central

C1C38 — C3×D76
C1C19C38C114C6×D19 — C3×D76
C19C38 — C3×D76
C1C6C12

Generators and relations for C3×D76
 G = < a,b,c | a3=b76=c2=1, ab=ba, ac=ca, cbc=b-1 >

38C2
38C2
19C22
19C22
38C6
38C6
2D19
2D19
19D4
19C2×C6
19C2×C6
2C3×D19
2C3×D19
19C3×D4

Smallest permutation representation of C3×D76
On 228 points
Generators in S228
(1 107 157)(2 108 158)(3 109 159)(4 110 160)(5 111 161)(6 112 162)(7 113 163)(8 114 164)(9 115 165)(10 116 166)(11 117 167)(12 118 168)(13 119 169)(14 120 170)(15 121 171)(16 122 172)(17 123 173)(18 124 174)(19 125 175)(20 126 176)(21 127 177)(22 128 178)(23 129 179)(24 130 180)(25 131 181)(26 132 182)(27 133 183)(28 134 184)(29 135 185)(30 136 186)(31 137 187)(32 138 188)(33 139 189)(34 140 190)(35 141 191)(36 142 192)(37 143 193)(38 144 194)(39 145 195)(40 146 196)(41 147 197)(42 148 198)(43 149 199)(44 150 200)(45 151 201)(46 152 202)(47 77 203)(48 78 204)(49 79 205)(50 80 206)(51 81 207)(52 82 208)(53 83 209)(54 84 210)(55 85 211)(56 86 212)(57 87 213)(58 88 214)(59 89 215)(60 90 216)(61 91 217)(62 92 218)(63 93 219)(64 94 220)(65 95 221)(66 96 222)(67 97 223)(68 98 224)(69 99 225)(70 100 226)(71 101 227)(72 102 228)(73 103 153)(74 104 154)(75 105 155)(76 106 156)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(77 79)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 144)(89 143)(90 142)(91 141)(92 140)(93 139)(94 138)(95 137)(96 136)(97 135)(98 134)(99 133)(100 132)(101 131)(102 130)(103 129)(104 128)(105 127)(106 126)(107 125)(108 124)(109 123)(110 122)(111 121)(112 120)(113 119)(114 118)(115 117)(153 179)(154 178)(155 177)(156 176)(157 175)(158 174)(159 173)(160 172)(161 171)(162 170)(163 169)(164 168)(165 167)(180 228)(181 227)(182 226)(183 225)(184 224)(185 223)(186 222)(187 221)(188 220)(189 219)(190 218)(191 217)(192 216)(193 215)(194 214)(195 213)(196 212)(197 211)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)

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

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

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

123 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F12A12B19A···19I38A···38I57A···57R76A···76R114A···114R228A···228AJ
order1222334666666121219···1938···3857···5776···76114···114228···228
size1138381121138383838222···22···22···22···22···22···2

123 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4C3×D4D19D38C3×D19D76C6×D19C3×D76
kernelC3×D76C228C6×D19D76C76D38C57C19C12C6C4C3C2C1
# reps112224129918181836

Matrix representation of C3×D76 in GL3(𝔽229) generated by

9400
010
001
,
22800
036110
0119141
,
22800
0134104
010795
G:=sub<GL(3,GF(229))| [94,0,0,0,1,0,0,0,1],[228,0,0,0,36,119,0,110,141],[228,0,0,0,134,107,0,104,95] >;

C3×D76 in GAP, Magma, Sage, TeX

C_3\times D_{76}
% in TeX

G:=Group("C3xD76");
// GroupNames label

G:=SmallGroup(456,26);
// by ID

G=gap.SmallGroup(456,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,141,66,10804]);
// Polycyclic

G:=Group<a,b,c|a^3=b^76=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D76 in TeX

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