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G = S3×C76order 456 = 23·3·19

Direct product of C76 and S3

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

Aliases: S3×C76, D6.C38, C2286C2, C122C38, C38.14D6, Dic32C38, C114.19C22, C31(C2×C76), C576(C2×C4), C2.1(S3×C38), C6.2(C2×C38), (S3×C38).2C2, (Dic3×C19)⋊5C2, SmallGroup(456,30)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C76
C1C3C6C114S3×C38 — S3×C76
C3 — S3×C76
C1C76

Generators and relations for S3×C76
 G = < a,b,c | a76=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C4
3C22
3C38
3C38
3C2×C4
3C76
3C2×C38
3C2×C76

Smallest permutation representation of S3×C76
On 228 points
Generators in S228
(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)
(1 179 122)(2 180 123)(3 181 124)(4 182 125)(5 183 126)(6 184 127)(7 185 128)(8 186 129)(9 187 130)(10 188 131)(11 189 132)(12 190 133)(13 191 134)(14 192 135)(15 193 136)(16 194 137)(17 195 138)(18 196 139)(19 197 140)(20 198 141)(21 199 142)(22 200 143)(23 201 144)(24 202 145)(25 203 146)(26 204 147)(27 205 148)(28 206 149)(29 207 150)(30 208 151)(31 209 152)(32 210 77)(33 211 78)(34 212 79)(35 213 80)(36 214 81)(37 215 82)(38 216 83)(39 217 84)(40 218 85)(41 219 86)(42 220 87)(43 221 88)(44 222 89)(45 223 90)(46 224 91)(47 225 92)(48 226 93)(49 227 94)(50 228 95)(51 153 96)(52 154 97)(53 155 98)(54 156 99)(55 157 100)(56 158 101)(57 159 102)(58 160 103)(59 161 104)(60 162 105)(61 163 106)(62 164 107)(63 165 108)(64 166 109)(65 167 110)(66 168 111)(67 169 112)(68 170 113)(69 171 114)(70 172 115)(71 173 116)(72 174 117)(73 175 118)(74 176 119)(75 177 120)(76 178 121)
(77 210)(78 211)(79 212)(80 213)(81 214)(82 215)(83 216)(84 217)(85 218)(86 219)(87 220)(88 221)(89 222)(90 223)(91 224)(92 225)(93 226)(94 227)(95 228)(96 153)(97 154)(98 155)(99 156)(100 157)(101 158)(102 159)(103 160)(104 161)(105 162)(106 163)(107 164)(108 165)(109 166)(110 167)(111 168)(112 169)(113 170)(114 171)(115 172)(116 173)(117 174)(118 175)(119 176)(120 177)(121 178)(122 179)(123 180)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 193)(137 194)(138 195)(139 196)(140 197)(141 198)(142 199)(143 200)(144 201)(145 202)(146 203)(147 204)(148 205)(149 206)(150 207)(151 208)(152 209)

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

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

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

228 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B19A···19R38A···38R38S···38BB57A···57R76A···76AJ76AK···76BT114A···114R228A···228AJ
order1222344446121219···1938···3838···3857···5776···7676···76114···114228···228
size1133211332221···11···13···32···21···13···32···22···2

228 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C19C38C38C38C76S3D6C4×S3S3×C19S3×C38S3×C76
kernelS3×C76Dic3×C19C228S3×C38S3×C19C4×S3Dic3C12D6S3C76C38C19C4C2C1
# reps111141818181872112181836

Matrix representation of S3×C76 in GL3(𝔽229) generated by

10700
0610
0061
,
100
00228
01228
,
22800
001
010
G:=sub<GL(3,GF(229))| [107,0,0,0,61,0,0,0,61],[1,0,0,0,0,1,0,228,228],[228,0,0,0,0,1,0,1,0] >;

S3×C76 in GAP, Magma, Sage, TeX

S_3\times C_{76}
% in TeX

G:=Group("S3xC76");
// GroupNames label

G:=SmallGroup(456,30);
// by ID

G=gap.SmallGroup(456,30);
# by ID

G:=PCGroup([5,-2,-2,-19,-2,-3,386,7604]);
// Polycyclic

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

Export

Subgroup lattice of S3×C76 in TeX

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