metabelian, supersoluble, monomial
Aliases: C72⋊1S3, C24⋊1D9, C9⋊1D24, C3⋊1D72, C6.9D36, C18.9D12, C36.58D6, C12.58D18, C32.4D24, (C3×C9)⋊6D8, C8⋊1(C9⋊S3), (C3×C72)⋊3C2, C36⋊S3⋊2C2, C3.(C32⋊5D8), C24.2(C3⋊S3), (C3×C24).12S3, (C3×C18).31D4, (C3×C6).56D12, (C3×C12).193D6, C6.3(C12⋊S3), C2.5(C36⋊S3), (C3×C36).60C22, C4.10(C2×C9⋊S3), C12.61(C2×C3⋊S3), SmallGroup(432,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊1S3
G = < a,b,c | a72=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Subgroups: 1180 in 110 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C9, C32, C12, C12, D6, D8, D9, C18, C3⋊S3, C3×C6, C24, C24, D12, C3×C9, C36, D18, C3×C12, C2×C3⋊S3, D24, C9⋊S3, C3×C18, C72, D36, C3×C24, C12⋊S3, C3×C36, C2×C9⋊S3, D72, C32⋊5D8, C3×C72, C36⋊S3, C72⋊1S3
Quotients: C1, C2, C22, S3, D4, D6, D8, D9, C3⋊S3, D12, D18, C2×C3⋊S3, D24, C9⋊S3, D36, C12⋊S3, C2×C9⋊S3, D72, C32⋊5D8, C36⋊S3, C72⋊1S3
►On 216 points
Generators in S216
(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)
(1 168 103)(2 169 104)(3 170 105)(4 171 106)(5 172 107)(6 173 108)(7 174 109)(8 175 110)(9 176 111)(10 177 112)(11 178 113)(12 179 114)(13 180 115)(14 181 116)(15 182 117)(16 183 118)(17 184 119)(18 185 120)(19 186 121)(20 187 122)(21 188 123)(22 189 124)(23 190 125)(24 191 126)(25 192 127)(26 193 128)(27 194 129)(28 195 130)(29 196 131)(30 197 132)(31 198 133)(32 199 134)(33 200 135)(34 201 136)(35 202 137)(36 203 138)(37 204 139)(38 205 140)(39 206 141)(40 207 142)(41 208 143)(42 209 144)(43 210 73)(44 211 74)(45 212 75)(46 213 76)(47 214 77)(48 215 78)(49 216 79)(50 145 80)(51 146 81)(52 147 82)(53 148 83)(54 149 84)(55 150 85)(56 151 86)(57 152 87)(58 153 88)(59 154 89)(60 155 90)(61 156 91)(62 157 92)(63 158 93)(64 159 94)(65 160 95)(66 161 96)(67 162 97)(68 163 98)(69 164 99)(70 165 100)(71 166 101)(72 167 102)
(1 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(65 72)(66 71)(67 70)(68 69)(73 189)(74 188)(75 187)(76 186)(77 185)(78 184)(79 183)(80 182)(81 181)(82 180)(83 179)(84 178)(85 177)(86 176)(87 175)(88 174)(89 173)(90 172)(91 171)(92 170)(93 169)(94 168)(95 167)(96 166)(97 165)(98 164)(99 163)(100 162)(101 161)(102 160)(103 159)(104 158)(105 157)(106 156)(107 155)(108 154)(109 153)(110 152)(111 151)(112 150)(113 149)(114 148)(115 147)(116 146)(117 145)(118 216)(119 215)(120 214)(121 213)(122 212)(123 211)(124 210)(125 209)(126 208)(127 207)(128 206)(129 205)(130 204)(131 203)(132 202)(133 201)(134 200)(135 199)(136 198)(137 197)(138 196)(139 195)(140 194)(141 193)(142 192)(143 191)(144 190)
G:=sub<Sym(216)| (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), (1,168,103)(2,169,104)(3,170,105)(4,171,106)(5,172,107)(6,173,108)(7,174,109)(8,175,110)(9,176,111)(10,177,112)(11,178,113)(12,179,114)(13,180,115)(14,181,116)(15,182,117)(16,183,118)(17,184,119)(18,185,120)(19,186,121)(20,187,122)(21,188,123)(22,189,124)(23,190,125)(24,191,126)(25,192,127)(26,193,128)(27,194,129)(28,195,130)(29,196,131)(30,197,132)(31,198,133)(32,199,134)(33,200,135)(34,201,136)(35,202,137)(36,203,138)(37,204,139)(38,205,140)(39,206,141)(40,207,142)(41,208,143)(42,209,144)(43,210,73)(44,211,74)(45,212,75)(46,213,76)(47,214,77)(48,215,78)(49,216,79)(50,145,80)(51,146,81)(52,147,82)(53,148,83)(54,149,84)(55,150,85)(56,151,86)(57,152,87)(58,153,88)(59,154,89)(60,155,90)(61,156,91)(62,157,92)(63,158,93)(64,159,94)(65,160,95)(66,161,96)(67,162,97)(68,163,98)(69,164,99)(70,165,100)(71,166,101)(72,167,102), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(65,72)(66,71)(67,70)(68,69)(73,189)(74,188)(75,187)(76,186)(77,185)(78,184)(79,183)(80,182)(81,181)(82,180)(83,179)(84,178)(85,177)(86,176)(87,175)(88,174)(89,173)(90,172)(91,171)(92,170)(93,169)(94,168)(95,167)(96,166)(97,165)(98,164)(99,163)(100,162)(101,161)(102,160)(103,159)(104,158)(105,157)(106,156)(107,155)(108,154)(109,153)(110,152)(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,216)(119,215)(120,214)(121,213)(122,212)(123,211)(124,210)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)>;
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), (1,168,103)(2,169,104)(3,170,105)(4,171,106)(5,172,107)(6,173,108)(7,174,109)(8,175,110)(9,176,111)(10,177,112)(11,178,113)(12,179,114)(13,180,115)(14,181,116)(15,182,117)(16,183,118)(17,184,119)(18,185,120)(19,186,121)(20,187,122)(21,188,123)(22,189,124)(23,190,125)(24,191,126)(25,192,127)(26,193,128)(27,194,129)(28,195,130)(29,196,131)(30,197,132)(31,198,133)(32,199,134)(33,200,135)(34,201,136)(35,202,137)(36,203,138)(37,204,139)(38,205,140)(39,206,141)(40,207,142)(41,208,143)(42,209,144)(43,210,73)(44,211,74)(45,212,75)(46,213,76)(47,214,77)(48,215,78)(49,216,79)(50,145,80)(51,146,81)(52,147,82)(53,148,83)(54,149,84)(55,150,85)(56,151,86)(57,152,87)(58,153,88)(59,154,89)(60,155,90)(61,156,91)(62,157,92)(63,158,93)(64,159,94)(65,160,95)(66,161,96)(67,162,97)(68,163,98)(69,164,99)(70,165,100)(71,166,101)(72,167,102), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(65,72)(66,71)(67,70)(68,69)(73,189)(74,188)(75,187)(76,186)(77,185)(78,184)(79,183)(80,182)(81,181)(82,180)(83,179)(84,178)(85,177)(86,176)(87,175)(88,174)(89,173)(90,172)(91,171)(92,170)(93,169)(94,168)(95,167)(96,166)(97,165)(98,164)(99,163)(100,162)(101,161)(102,160)(103,159)(104,158)(105,157)(106,156)(107,155)(108,154)(109,153)(110,152)(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,216)(119,215)(120,214)(121,213)(122,212)(123,211)(124,210)(125,209)(126,208)(127,207)(128,206)(129,205)(130,204)(131,203)(132,202)(133,201)(134,200)(135,199)(136,198)(137,197)(138,196)(139,195)(140,194)(141,193)(142,192)(143,191)(144,190) );
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)], [(1,168,103),(2,169,104),(3,170,105),(4,171,106),(5,172,107),(6,173,108),(7,174,109),(8,175,110),(9,176,111),(10,177,112),(11,178,113),(12,179,114),(13,180,115),(14,181,116),(15,182,117),(16,183,118),(17,184,119),(18,185,120),(19,186,121),(20,187,122),(21,188,123),(22,189,124),(23,190,125),(24,191,126),(25,192,127),(26,193,128),(27,194,129),(28,195,130),(29,196,131),(30,197,132),(31,198,133),(32,199,134),(33,200,135),(34,201,136),(35,202,137),(36,203,138),(37,204,139),(38,205,140),(39,206,141),(40,207,142),(41,208,143),(42,209,144),(43,210,73),(44,211,74),(45,212,75),(46,213,76),(47,214,77),(48,215,78),(49,216,79),(50,145,80),(51,146,81),(52,147,82),(53,148,83),(54,149,84),(55,150,85),(56,151,86),(57,152,87),(58,153,88),(59,154,89),(60,155,90),(61,156,91),(62,157,92),(63,158,93),(64,159,94),(65,160,95),(66,161,96),(67,162,97),(68,163,98),(69,164,99),(70,165,100),(71,166,101),(72,167,102)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(65,72),(66,71),(67,70),(68,69),(73,189),(74,188),(75,187),(76,186),(77,185),(78,184),(79,183),(80,182),(81,181),(82,180),(83,179),(84,178),(85,177),(86,176),(87,175),(88,174),(89,173),(90,172),(91,171),(92,170),(93,169),(94,168),(95,167),(96,166),(97,165),(98,164),(99,163),(100,162),(101,161),(102,160),(103,159),(104,158),(105,157),(106,156),(107,155),(108,154),(109,153),(110,152),(111,151),(112,150),(113,149),(114,148),(115,147),(116,146),(117,145),(118,216),(119,215),(120,214),(121,213),(122,212),(123,211),(124,210),(125,209),(126,208),(127,207),(128,206),(129,205),(130,204),(131,203),(132,202),(133,201),(134,200),(135,199),(136,198),(137,197),(138,196),(139,195),(140,194),(141,193),(142,192),(143,191),(144,190)]])
111 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 8A | 8B | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 108 | 108 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D8 | D9 | D12 | D12 | D18 | D24 | D24 | D36 | D72 |
| kernel | C72⋊1S3 | C3×C72 | C36⋊S3 | C72 | C3×C24 | C3×C18 | C36 | C3×C12 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 2 | 9 | 6 | 2 | 9 | 12 | 4 | 18 | 36 |
Matrix representation of C72⋊1S3 ►in GL4(𝔽73) generated by
| 2 | 60 | 0 | 0 |
| 13 | 62 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 1 |
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 5 | 18 | 0 | 0 |
| 23 | 68 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(73))| [2,13,0,0,60,62,0,0,0,0,0,1,0,0,72,1],[0,1,0,0,72,72,0,0,0,0,72,1,0,0,72,0],[5,23,0,0,18,68,0,0,0,0,72,1,0,0,0,1] >;
C72⋊1S3 in GAP, Magma, Sage, TeX
C_{72}\rtimes_1S_3 % in TeX
G:=Group("C72:1S3"); // GroupNames label
G:=SmallGroup(432,172);
// by ID
G=gap.SmallGroup(432,172);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^72=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations