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## G = C3⋊S3×C23order 414 = 2·32·23

### Direct product of C23 and C3⋊S3

Aliases: C3⋊S3×C23, C693S3, C322C46, C3⋊(S3×C23), (C3×C69)⋊5C2, SmallGroup(414,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C23
 Chief series C1 — C3 — C32 — C3×C69 — C3⋊S3×C23
 Lower central C32 — C3⋊S3×C23
 Upper central C1 — C23

Generators and relations for C3⋊S3×C23
G = < a,b,c,d | a23=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Smallest permutation representation of C3⋊S3×C23
On 207 points
Generators in S207
(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)
(1 38 80)(2 39 81)(3 40 82)(4 41 83)(5 42 84)(6 43 85)(7 44 86)(8 45 87)(9 46 88)(10 24 89)(11 25 90)(12 26 91)(13 27 92)(14 28 70)(15 29 71)(16 30 72)(17 31 73)(18 32 74)(19 33 75)(20 34 76)(21 35 77)(22 36 78)(23 37 79)(47 136 161)(48 137 139)(49 138 140)(50 116 141)(51 117 142)(52 118 143)(53 119 144)(54 120 145)(55 121 146)(56 122 147)(57 123 148)(58 124 149)(59 125 150)(60 126 151)(61 127 152)(62 128 153)(63 129 154)(64 130 155)(65 131 156)(66 132 157)(67 133 158)(68 134 159)(69 135 160)(93 201 164)(94 202 165)(95 203 166)(96 204 167)(97 205 168)(98 206 169)(99 207 170)(100 185 171)(101 186 172)(102 187 173)(103 188 174)(104 189 175)(105 190 176)(106 191 177)(107 192 178)(108 193 179)(109 194 180)(110 195 181)(111 196 182)(112 197 183)(113 198 184)(114 199 162)(115 200 163)
(1 171 146)(2 172 147)(3 173 148)(4 174 149)(5 175 150)(6 176 151)(7 177 152)(8 178 153)(9 179 154)(10 180 155)(11 181 156)(12 182 157)(13 183 158)(14 184 159)(15 162 160)(16 163 161)(17 164 139)(18 165 140)(19 166 141)(20 167 142)(21 168 143)(22 169 144)(23 170 145)(24 109 64)(25 110 65)(26 111 66)(27 112 67)(28 113 68)(29 114 69)(30 115 47)(31 93 48)(32 94 49)(33 95 50)(34 96 51)(35 97 52)(36 98 53)(37 99 54)(38 100 55)(39 101 56)(40 102 57)(41 103 58)(42 104 59)(43 105 60)(44 106 61)(45 107 62)(46 108 63)(70 198 134)(71 199 135)(72 200 136)(73 201 137)(74 202 138)(75 203 116)(76 204 117)(77 205 118)(78 206 119)(79 207 120)(80 185 121)(81 186 122)(82 187 123)(83 188 124)(84 189 125)(85 190 126)(86 191 127)(87 192 128)(88 193 129)(89 194 130)(90 195 131)(91 196 132)(92 197 133)
(24 89)(25 90)(26 91)(27 92)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 200)(48 201)(49 202)(50 203)(51 204)(52 205)(53 206)(54 207)(55 185)(56 186)(57 187)(58 188)(59 189)(60 190)(61 191)(62 192)(63 193)(64 194)(65 195)(66 196)(67 197)(68 198)(69 199)(93 137)(94 138)(95 116)(96 117)(97 118)(98 119)(99 120)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 136)(139 164)(140 165)(141 166)(142 167)(143 168)(144 169)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 181)(157 182)(158 183)(159 184)(160 162)(161 163)

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

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

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

138 conjugacy classes

 class 1 2 3A 3B 3C 3D 23A ··· 23V 46A ··· 46V 69A ··· 69CJ order 1 2 3 3 3 3 23 ··· 23 46 ··· 46 69 ··· 69 size 1 9 2 2 2 2 1 ··· 1 9 ··· 9 2 ··· 2

138 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C23 C46 S3 S3×C23 kernel C3⋊S3×C23 C3×C69 C3⋊S3 C32 C69 C3 # reps 1 1 22 22 4 88

Matrix representation of C3⋊S3×C23 in GL4(𝔽139) generated by

 77 0 0 0 0 77 0 0 0 0 77 0 0 0 0 77
,
 138 1 0 0 138 0 0 0 0 0 1 0 0 0 0 1
,
 138 1 0 0 138 0 0 0 0 0 137 136 0 0 1 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 138 138
G:=sub<GL(4,GF(139))| [77,0,0,0,0,77,0,0,0,0,77,0,0,0,0,77],[138,138,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[138,138,0,0,1,0,0,0,0,0,137,1,0,0,136,1],[0,1,0,0,1,0,0,0,0,0,1,138,0,0,0,138] >;

C3⋊S3×C23 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_{23}
% in TeX

G:=Group("C3:S3xC23");
// GroupNames label

G:=SmallGroup(414,8);
// by ID

G=gap.SmallGroup(414,8);
# by ID

G:=PCGroup([4,-2,-23,-3,-3,1106,4419]);
// Polycyclic

G:=Group<a,b,c,d|a^23=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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