metabelian, supersoluble, monomial
Aliases: C72⋊4S3, C24⋊5D9, C36.67D6, C12.67D18, C8⋊3(C9⋊S3), (C3×C72)⋊9C2, C6.12(C4×D9), C9⋊2(C8⋊S3), (C3×C9)⋊6M4(2), C3.(C24⋊S3), C3⋊2(C8⋊D9), C18.13(C4×S3), (C3×C24).24S3, C9⋊Dic3.3C4, C24.10(C3⋊S3), (C3×C12).217D6, C36.S3⋊11C2, (C3×C36).70C22, C32.5(C8⋊S3), C2.3(C4×C9⋊S3), C6.7(C4×C3⋊S3), (C2×C9⋊S3).3C4, (C4×C9⋊S3).5C2, C4.13(C2×C9⋊S3), C12.69(C2×C3⋊S3), (C3×C6).69(C4×S3), (C3×C18).24(C2×C4), SmallGroup(432,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊S3
G = < a,b,c | a72=b3=c2=1, ab=ba, cac=a53, cbc=b-1 >
Subgroups: 572 in 100 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C32, Dic3, C12, C12, D6, M4(2), D9, C18, C3⋊S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C3×C9, Dic9, C36, D18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, C9⋊S3, C3×C18, C9⋊C8, C72, C4×D9, C32⋊4C8, C3×C24, C4×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C8⋊D9, C24⋊S3, C36.S3, C3×C72, C4×C9⋊S3, C72⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), D9, C3⋊S3, C4×S3, D18, C2×C3⋊S3, C8⋊S3, C9⋊S3, C4×D9, C4×C3⋊S3, C2×C9⋊S3, C8⋊D9, C24⋊S3, C4×C9⋊S3, C72⋊S3
►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)
(1 158 127)(2 159 128)(3 160 129)(4 161 130)(5 162 131)(6 163 132)(7 164 133)(8 165 134)(9 166 135)(10 167 136)(11 168 137)(12 169 138)(13 170 139)(14 171 140)(15 172 141)(16 173 142)(17 174 143)(18 175 144)(19 176 73)(20 177 74)(21 178 75)(22 179 76)(23 180 77)(24 181 78)(25 182 79)(26 183 80)(27 184 81)(28 185 82)(29 186 83)(30 187 84)(31 188 85)(32 189 86)(33 190 87)(34 191 88)(35 192 89)(36 193 90)(37 194 91)(38 195 92)(39 196 93)(40 197 94)(41 198 95)(42 199 96)(43 200 97)(44 201 98)(45 202 99)(46 203 100)(47 204 101)(48 205 102)(49 206 103)(50 207 104)(51 208 105)(52 209 106)(53 210 107)(54 211 108)(55 212 109)(56 213 110)(57 214 111)(58 215 112)(59 216 113)(60 145 114)(61 146 115)(62 147 116)(63 148 117)(64 149 118)(65 150 119)(66 151 120)(67 152 121)(68 153 122)(69 154 123)(70 155 124)(71 156 125)(72 157 126)
(2 54)(3 35)(4 16)(5 69)(6 50)(7 31)(8 12)(9 65)(10 46)(11 27)(13 61)(14 42)(15 23)(17 57)(18 38)(20 72)(21 53)(22 34)(24 68)(25 49)(26 30)(28 64)(29 45)(32 60)(33 41)(36 56)(39 71)(40 52)(43 67)(44 48)(47 63)(51 59)(58 70)(62 66)(73 176)(74 157)(75 210)(76 191)(77 172)(78 153)(79 206)(80 187)(81 168)(82 149)(83 202)(84 183)(85 164)(86 145)(87 198)(88 179)(89 160)(90 213)(91 194)(92 175)(93 156)(94 209)(95 190)(96 171)(97 152)(98 205)(99 186)(100 167)(101 148)(102 201)(103 182)(104 163)(105 216)(106 197)(107 178)(108 159)(109 212)(110 193)(111 174)(112 155)(113 208)(114 189)(115 170)(116 151)(117 204)(118 185)(119 166)(120 147)(121 200)(122 181)(123 162)(124 215)(125 196)(126 177)(127 158)(128 211)(129 192)(130 173)(131 154)(132 207)(133 188)(134 169)(135 150)(136 203)(137 184)(138 165)(139 146)(140 199)(141 180)(142 161)(143 214)(144 195)
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), (1,158,127)(2,159,128)(3,160,129)(4,161,130)(5,162,131)(6,163,132)(7,164,133)(8,165,134)(9,166,135)(10,167,136)(11,168,137)(12,169,138)(13,170,139)(14,171,140)(15,172,141)(16,173,142)(17,174,143)(18,175,144)(19,176,73)(20,177,74)(21,178,75)(22,179,76)(23,180,77)(24,181,78)(25,182,79)(26,183,80)(27,184,81)(28,185,82)(29,186,83)(30,187,84)(31,188,85)(32,189,86)(33,190,87)(34,191,88)(35,192,89)(36,193,90)(37,194,91)(38,195,92)(39,196,93)(40,197,94)(41,198,95)(42,199,96)(43,200,97)(44,201,98)(45,202,99)(46,203,100)(47,204,101)(48,205,102)(49,206,103)(50,207,104)(51,208,105)(52,209,106)(53,210,107)(54,211,108)(55,212,109)(56,213,110)(57,214,111)(58,215,112)(59,216,113)(60,145,114)(61,146,115)(62,147,116)(63,148,117)(64,149,118)(65,150,119)(66,151,120)(67,152,121)(68,153,122)(69,154,123)(70,155,124)(71,156,125)(72,157,126), (2,54)(3,35)(4,16)(5,69)(6,50)(7,31)(8,12)(9,65)(10,46)(11,27)(13,61)(14,42)(15,23)(17,57)(18,38)(20,72)(21,53)(22,34)(24,68)(25,49)(26,30)(28,64)(29,45)(32,60)(33,41)(36,56)(39,71)(40,52)(43,67)(44,48)(47,63)(51,59)(58,70)(62,66)(73,176)(74,157)(75,210)(76,191)(77,172)(78,153)(79,206)(80,187)(81,168)(82,149)(83,202)(84,183)(85,164)(86,145)(87,198)(88,179)(89,160)(90,213)(91,194)(92,175)(93,156)(94,209)(95,190)(96,171)(97,152)(98,205)(99,186)(100,167)(101,148)(102,201)(103,182)(104,163)(105,216)(106,197)(107,178)(108,159)(109,212)(110,193)(111,174)(112,155)(113,208)(114,189)(115,170)(116,151)(117,204)(118,185)(119,166)(120,147)(121,200)(122,181)(123,162)(124,215)(125,196)(126,177)(127,158)(128,211)(129,192)(130,173)(131,154)(132,207)(133,188)(134,169)(135,150)(136,203)(137,184)(138,165)(139,146)(140,199)(141,180)(142,161)(143,214)(144,195)>;
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), (1,158,127)(2,159,128)(3,160,129)(4,161,130)(5,162,131)(6,163,132)(7,164,133)(8,165,134)(9,166,135)(10,167,136)(11,168,137)(12,169,138)(13,170,139)(14,171,140)(15,172,141)(16,173,142)(17,174,143)(18,175,144)(19,176,73)(20,177,74)(21,178,75)(22,179,76)(23,180,77)(24,181,78)(25,182,79)(26,183,80)(27,184,81)(28,185,82)(29,186,83)(30,187,84)(31,188,85)(32,189,86)(33,190,87)(34,191,88)(35,192,89)(36,193,90)(37,194,91)(38,195,92)(39,196,93)(40,197,94)(41,198,95)(42,199,96)(43,200,97)(44,201,98)(45,202,99)(46,203,100)(47,204,101)(48,205,102)(49,206,103)(50,207,104)(51,208,105)(52,209,106)(53,210,107)(54,211,108)(55,212,109)(56,213,110)(57,214,111)(58,215,112)(59,216,113)(60,145,114)(61,146,115)(62,147,116)(63,148,117)(64,149,118)(65,150,119)(66,151,120)(67,152,121)(68,153,122)(69,154,123)(70,155,124)(71,156,125)(72,157,126), (2,54)(3,35)(4,16)(5,69)(6,50)(7,31)(8,12)(9,65)(10,46)(11,27)(13,61)(14,42)(15,23)(17,57)(18,38)(20,72)(21,53)(22,34)(24,68)(25,49)(26,30)(28,64)(29,45)(32,60)(33,41)(36,56)(39,71)(40,52)(43,67)(44,48)(47,63)(51,59)(58,70)(62,66)(73,176)(74,157)(75,210)(76,191)(77,172)(78,153)(79,206)(80,187)(81,168)(82,149)(83,202)(84,183)(85,164)(86,145)(87,198)(88,179)(89,160)(90,213)(91,194)(92,175)(93,156)(94,209)(95,190)(96,171)(97,152)(98,205)(99,186)(100,167)(101,148)(102,201)(103,182)(104,163)(105,216)(106,197)(107,178)(108,159)(109,212)(110,193)(111,174)(112,155)(113,208)(114,189)(115,170)(116,151)(117,204)(118,185)(119,166)(120,147)(121,200)(122,181)(123,162)(124,215)(125,196)(126,177)(127,158)(128,211)(129,192)(130,173)(131,154)(132,207)(133,188)(134,169)(135,150)(136,203)(137,184)(138,165)(139,146)(140,199)(141,180)(142,161)(143,214)(144,195) );
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)], [(1,158,127),(2,159,128),(3,160,129),(4,161,130),(5,162,131),(6,163,132),(7,164,133),(8,165,134),(9,166,135),(10,167,136),(11,168,137),(12,169,138),(13,170,139),(14,171,140),(15,172,141),(16,173,142),(17,174,143),(18,175,144),(19,176,73),(20,177,74),(21,178,75),(22,179,76),(23,180,77),(24,181,78),(25,182,79),(26,183,80),(27,184,81),(28,185,82),(29,186,83),(30,187,84),(31,188,85),(32,189,86),(33,190,87),(34,191,88),(35,192,89),(36,193,90),(37,194,91),(38,195,92),(39,196,93),(40,197,94),(41,198,95),(42,199,96),(43,200,97),(44,201,98),(45,202,99),(46,203,100),(47,204,101),(48,205,102),(49,206,103),(50,207,104),(51,208,105),(52,209,106),(53,210,107),(54,211,108),(55,212,109),(56,213,110),(57,214,111),(58,215,112),(59,216,113),(60,145,114),(61,146,115),(62,147,116),(63,148,117),(64,149,118),(65,150,119),(66,151,120),(67,152,121),(68,153,122),(69,154,123),(70,155,124),(71,156,125),(72,157,126)], [(2,54),(3,35),(4,16),(5,69),(6,50),(7,31),(8,12),(9,65),(10,46),(11,27),(13,61),(14,42),(15,23),(17,57),(18,38),(20,72),(21,53),(22,34),(24,68),(25,49),(26,30),(28,64),(29,45),(32,60),(33,41),(36,56),(39,71),(40,52),(43,67),(44,48),(47,63),(51,59),(58,70),(62,66),(73,176),(74,157),(75,210),(76,191),(77,172),(78,153),(79,206),(80,187),(81,168),(82,149),(83,202),(84,183),(85,164),(86,145),(87,198),(88,179),(89,160),(90,213),(91,194),(92,175),(93,156),(94,209),(95,190),(96,171),(97,152),(98,205),(99,186),(100,167),(101,148),(102,201),(103,182),(104,163),(105,216),(106,197),(107,178),(108,159),(109,212),(110,193),(111,174),(112,155),(113,208),(114,189),(115,170),(116,151),(117,204),(118,185),(119,166),(120,147),(121,200),(122,181),(123,162),(124,215),(125,196),(126,177),(127,158),(128,211),(129,192),(130,173),(131,154),(132,207),(133,188),(134,169),(135,150),(136,203),(137,184),(138,165),(139,146),(140,199),(141,180),(142,161),(143,214),(144,195)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18I | 24A | ··· | 24P | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 54 | 2 | 2 | 2 | 2 | 1 | 1 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 54 | 54 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | S3 | D6 | D6 | M4(2) | D9 | C4×S3 | C4×S3 | D18 | C8⋊S3 | C8⋊S3 | C4×D9 | C8⋊D9 |
| kernel | C72⋊S3 | C36.S3 | C3×C72 | C4×C9⋊S3 | C9⋊Dic3 | C2×C9⋊S3 | C72 | C3×C24 | C36 | C3×C12 | C3×C9 | C24 | C18 | C3×C6 | C12 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 1 | 3 | 1 | 2 | 9 | 6 | 2 | 9 | 12 | 4 | 18 | 36 |
Matrix representation of C72⋊S3 ►in GL4(𝔽73) generated by
| 39 | 65 | 0 | 0 |
| 8 | 47 | 0 | 0 |
| 0 | 0 | 2 | 29 |
| 0 | 0 | 44 | 31 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [39,8,0,0,65,47,0,0,0,0,2,44,0,0,29,31],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,0,72,0,0,72,0] >;
C72⋊S3 in GAP, Magma, Sage, TeX
C_{72}\rtimes S_3 % in TeX
G:=Group("C72:S3"); // GroupNames label
G:=SmallGroup(432,170);
// by ID
G=gap.SmallGroup(432,170);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,6164,662,4037,14118]);
// Polycyclic
G:=Group<a,b,c|a^72=b^3=c^2=1,a*b=b*a,c*a*c=a^53,c*b*c=b^-1>;
// generators/relations