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G = C62.148D6order 432 = 24·33

32nd non-split extension by C62 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: C62.148D6, (C6×C12)⋊11S3, (C3×C6).70D12, (C32×C6).79D4, C3213(D6⋊C4), C3316(C22⋊C4), C6.11(C12⋊S3), C32(C6.11D12), C6.23(C327D4), C2.2(C3315D4), C2.2(C3312D4), (C3×C62).54C22, (C3×C6×C12)⋊3C2, C6.16(C4×C3⋊S3), (C2×C12)⋊2(C3⋊S3), (C3×C6).84(C4×S3), (C2×C33⋊C2)⋊4C4, (C2×C335C4)⋊3C2, C2.5(C4×C33⋊C2), (C2×C4)⋊1(C33⋊C2), (C32×C6).63(C2×C4), (C3×C6).114(C3⋊D4), C22.6(C2×C33⋊C2), (C22×C33⋊C2).2C2, (C2×C6).43(C2×C3⋊S3), SmallGroup(432,506)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C62.148D6
C1C3C32C33C32×C6C3×C62C22×C33⋊C2 — C62.148D6
C33C32×C6 — C62.148D6
C1C22C2×C4

Generators and relations for C62.148D6
 G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 2984 in 476 conjugacy classes, 173 normal (15 characteristic)
C1, C2 [×3], C2 [×2], C3 [×13], C4 [×2], C22, C22 [×4], S3 [×26], C6 [×39], C2×C4, C2×C4, C23, C32 [×13], Dic3 [×13], C12 [×13], D6 [×52], C2×C6 [×13], C22⋊C4, C3⋊S3 [×26], C3×C6 [×39], C2×Dic3 [×13], C2×C12 [×13], C22×S3 [×13], C33, C3⋊Dic3 [×13], C3×C12 [×13], C2×C3⋊S3 [×52], C62 [×13], D6⋊C4 [×13], C33⋊C2 [×2], C32×C6 [×3], C2×C3⋊Dic3 [×13], C6×C12 [×13], C22×C3⋊S3 [×13], C335C4, C32×C12, C2×C33⋊C2 [×2], C2×C33⋊C2 [×2], C3×C62, C6.11D12 [×13], C2×C335C4, C3×C6×C12, C22×C33⋊C2, C62.148D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×13], C2×C4, D4 [×2], D6 [×13], C22⋊C4, C3⋊S3 [×13], C4×S3 [×13], D12 [×13], C3⋊D4 [×13], C2×C3⋊S3 [×13], D6⋊C4 [×13], C33⋊C2, C4×C3⋊S3 [×13], C12⋊S3 [×13], C327D4 [×13], C2×C33⋊C2, C6.11D12 [×13], C4×C33⋊C2, C3312D4, C3315D4, C62.148D6

Smallest permutation representation of C62.148D6
On 216 points
Generators in S216
(1 129 39 7 123 45)(2 130 40 8 124 46)(3 131 41 9 125 47)(4 132 42 10 126 48)(5 121 43 11 127 37)(6 122 44 12 128 38)(13 181 163 19 187 157)(14 182 164 20 188 158)(15 183 165 21 189 159)(16 184 166 22 190 160)(17 185 167 23 191 161)(18 186 168 24 192 162)(25 66 136 31 72 142)(26 67 137 32 61 143)(27 68 138 33 62 144)(28 69 139 34 63 133)(29 70 140 35 64 134)(30 71 141 36 65 135)(49 149 193 55 155 199)(50 150 194 56 156 200)(51 151 195 57 145 201)(52 152 196 58 146 202)(53 153 197 59 147 203)(54 154 198 60 148 204)(73 176 213 79 170 207)(74 177 214 80 171 208)(75 178 215 81 172 209)(76 179 216 82 173 210)(77 180 205 83 174 211)(78 169 206 84 175 212)(85 101 115 91 107 109)(86 102 116 92 108 110)(87 103 117 93 97 111)(88 104 118 94 98 112)(89 105 119 95 99 113)(90 106 120 96 100 114)
(1 216 163 101 35 200)(2 205 164 102 36 201)(3 206 165 103 25 202)(4 207 166 104 26 203)(5 208 167 105 27 204)(6 209 168 106 28 193)(7 210 157 107 29 194)(8 211 158 108 30 195)(9 212 159 97 31 196)(10 213 160 98 32 197)(11 214 161 99 33 198)(12 215 162 100 34 199)(13 109 70 56 123 76)(14 110 71 57 124 77)(15 111 72 58 125 78)(16 112 61 59 126 79)(17 113 62 60 127 80)(18 114 63 49 128 81)(19 115 64 50 129 82)(20 116 65 51 130 83)(21 117 66 52 131 84)(22 118 67 53 132 73)(23 119 68 54 121 74)(24 120 69 55 122 75)(37 171 185 89 144 148)(38 172 186 90 133 149)(39 173 187 91 134 150)(40 174 188 92 135 151)(41 175 189 93 136 152)(42 176 190 94 137 153)(43 177 191 95 138 154)(44 178 192 96 139 155)(45 179 181 85 140 156)(46 180 182 86 141 145)(47 169 183 87 142 146)(48 170 184 88 143 147)
(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 12 107 106)(2 105 108 11)(3 10 97 104)(4 103 98 9)(5 8 99 102)(6 101 100 7)(13 133 50 178)(14 177 51 144)(15 143 52 176)(16 175 53 142)(17 141 54 174)(18 173 55 140)(19 139 56 172)(20 171 57 138)(21 137 58 170)(22 169 59 136)(23 135 60 180)(24 179 49 134)(25 160 212 203)(26 202 213 159)(27 158 214 201)(28 200 215 157)(29 168 216 199)(30 198 205 167)(31 166 206 197)(32 196 207 165)(33 164 208 195)(34 194 209 163)(35 162 210 193)(36 204 211 161)(37 124 95 116)(38 115 96 123)(39 122 85 114)(40 113 86 121)(41 132 87 112)(42 111 88 131)(43 130 89 110)(44 109 90 129)(45 128 91 120)(46 119 92 127)(47 126 93 118)(48 117 94 125)(61 152 73 183)(62 182 74 151)(63 150 75 181)(64 192 76 149)(65 148 77 191)(66 190 78 147)(67 146 79 189)(68 188 80 145)(69 156 81 187)(70 186 82 155)(71 154 83 185)(72 184 84 153)

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

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

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

114 conjugacy classes

class 1 2A2B2C2D2E3A···3M4A4B4C4D6A···6AM12A···12AZ
order1222223···344446···612···12
size111154542···22254542···22···2

114 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4S3D4D6C4×S3D12C3⋊D4
kernelC62.148D6C2×C335C4C3×C6×C12C22×C33⋊C2C2×C33⋊C2C6×C12C32×C6C62C3×C6C3×C6C3×C6
# reps1111413213262626

Matrix representation of C62.148D6 in GL6(𝔽13)

12100000
120000
000100
0012100
0000012
0000112
,
230000
12120000
0012000
0001200
0000112
000010
,
440000
300000
003300
0010600
000022
0000114
,
440000
690000
0010600
003300
00001111
000092

G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,10,2,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[2,12,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[4,3,0,0,0,0,4,0,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,2,11,0,0,0,0,2,4],[4,6,0,0,0,0,4,9,0,0,0,0,0,0,10,3,0,0,0,0,6,3,0,0,0,0,0,0,11,9,0,0,0,0,11,2] >;

C62.148D6 in GAP, Magma, Sage, TeX

C_6^2._{148}D_6
% in TeX

G:=Group("C6^2.148D6");
// GroupNames label

G:=SmallGroup(432,506);
// by ID

G=gap.SmallGroup(432,506);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,1124,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,d^2=a^3*b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
// generators/relations

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