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G = C62.100D6  order 432 = 24·33

48th non-split extension by C62 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C62.100D6, (D4×C33)⋊9C2, C33⋊8Q8⋊9C2, (C3×C12).152D6, (D4×C32)⋊12S3, C33⋊32(C4○D4), C33⋊15D4⋊6C2, D4⋊2(C33⋊C2), C3⋊5(C12.D6), (C3×C62).39C22, C32⋊27(D4⋊2S3), (C32×C6).100C23, (C32×C12).57C22, C33⋊5C4.21C22, (C3×D4)⋊3(C3⋊S3), C12.27(C2×C3⋊S3), (C4×C33⋊C2)⋊5C2, (C2×C33⋊5C4)⋊9C2, C6.44(C22×C3⋊S3), C4.5(C2×C33⋊C2), (C3×C6).189(C22×S3), C22.1(C2×C33⋊C2), C2.7(C22×C33⋊C2), (C2×C33⋊C2).19C22, (C2×C6).8(C2×C3⋊S3), SmallGroup(432,725)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C62.100D6
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C2×C33⋊C2 — C4×C33⋊C2 — C62.100D6
Lower central
C33 — C32×C6 — C62.100D6
Upper central
C1 — C2 — D4

Generators and relations for C62.100D6
 G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 2672 in 560 conjugacy classes, 179 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C33, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, D4⋊2S3, C33⋊C2, C32×C6, C32×C6, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C33⋊5C4, C33⋊5C4, C32×C12, C2×C33⋊C2, C3×C62, C12.D6, C33⋊8Q8, C4×C33⋊C2, C2×C33⋊5C4, C33⋊15D4, D4×C33, C62.100D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, D4⋊2S3, C33⋊C2, C22×C3⋊S3, C2×C33⋊C2, C12.D6, C22×C33⋊C2, C62.100D6

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

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

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

75 conjugacy classes

class 1 2A2B2C2D3A···3M4A4B4C4D4E6A···6M6N···6AM12A···12M
order122223···3444446···66···612···12
size1122542···22272754542···24···44···4

75 irreducible representations

dim11111122224
type+++++++++-
imageC1C2C2C2C2C2S3D6D6C4○D4D4⋊2S3
kernelC62.100D6C33⋊8Q8C4×C33⋊C2C2×C33⋊5C4C33⋊15D4D4×C33D4×C32C3×C12C62C33C32
# reps111221131326213

Matrix representation of C62.100D6 ►in GL8(𝔽13)

11000000
120000000
00400000
0012100000
000001200
000011200
00000018
000000012
,
10000000
01000000
00100000
00010000
000001200
000011200
000000120
000000012
,
012000000
11000000
00400000
0012100000
000001200
000011200
00000018
000000312
,
01000000
10000000
001050000
00130000
000011200
000001200
00000080
00000008

G:=sub<GL(8,GF(13))| [1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,12,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,4,12,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,8,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,5,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8] >;
 

C62.100D6 in GAP, Magma, Sage, TeX 

C_6^2._{100}D_6
 
% in TeX
 
G:=Group("C6^2.100D6");
 
// GroupNames label
 
G:=SmallGroup(432,725);
 
// by ID
 
G=gap.SmallGroup(432,725);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,1124,4037,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
 
// generators/relations
 

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