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## G = C63.C2order 432 = 24·33

### 4th non-split extension by C63 of C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C63.C2
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — C2×C33⋊5C4 — C63.C2
 Lower central C33 — C32×C6 — C63.C2
 Upper central C1 — C22 — C23

Generators and relations for C63.C2
G = < a,b,c,d | a6=b6=c6=1, d2=b3, ab=ba, ac=ca, dad-1=a-1c3, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 1736 in 476 conjugacy classes, 227 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C23, C32, Dic3, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×Dic3, C22×C6, C33, C3⋊Dic3, C62, C62, C6.D4, C32×C6, C32×C6, C32×C6, C2×C3⋊Dic3, C2×C62, C335C4, C3×C62, C3×C62, C3×C62, C625C4, C2×C335C4, C63, C63.C2
Quotients:

Smallest permutation representation of C63.C2
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 80 18 4 83 15)(2 81 13 5 84 16)(3 82 14 6 79 17)(7 198 58 111 147 192)(8 193 59 112 148 187)(9 194 60 113 149 188)(10 195 55 114 150 189)(11 196 56 109 145 190)(12 197 57 110 146 191)(19 120 62 211 77 123)(20 115 63 212 78 124)(21 116 64 213 73 125)(22 117 65 214 74 126)(23 118 66 215 75 121)(24 119 61 216 76 122)(25 90 51 28 87 54)(26 85 52 29 88 49)(27 86 53 30 89 50)(31 43 105 34 46 108)(32 44 106 35 47 103)(33 45 107 36 48 104)(37 98 67 40 101 70)(38 99 68 41 102 71)(39 100 69 42 97 72)(91 172 151 94 169 154)(92 173 152 95 170 155)(93 174 153 96 171 156)(127 136 184 201 142 206)(128 137 185 202 143 207)(129 138 186 203 144 208)(130 133 181 204 139 209)(131 134 182 199 140 210)(132 135 183 200 141 205)(157 165 177 160 168 180)(158 166 178 161 163 175)(159 167 179 162 164 176)
(1 152 44 53 40 178)(2 153 45 54 41 179)(3 154 46 49 42 180)(4 155 47 50 37 175)(5 156 48 51 38 176)(6 151 43 52 39 177)(7 199 65 114 128 123)(8 200 66 109 129 124)(9 201 61 110 130 125)(10 202 62 111 131 126)(11 203 63 112 132 121)(12 204 64 113 127 122)(13 171 36 90 71 164)(14 172 31 85 72 165)(15 173 32 86 67 166)(16 174 33 87 68 167)(17 169 34 88 69 168)(18 170 35 89 70 163)(19 198 140 214 150 137)(20 193 141 215 145 138)(21 194 142 216 146 133)(22 195 143 211 147 134)(23 196 144 212 148 135)(24 197 139 213 149 136)(25 102 162 81 96 107)(26 97 157 82 91 108)(27 98 158 83 92 103)(28 99 159 84 93 104)(29 100 160 79 94 105)(30 101 161 80 95 106)(55 207 77 192 182 117)(56 208 78 187 183 118)(57 209 73 188 184 119)(58 210 74 189 185 120)(59 205 75 190 186 115)(60 206 76 191 181 116)
(1 205 4 183)(2 185 5 207)(3 209 6 181)(7 68 111 71)(8 173 112 170)(9 72 113 69)(10 171 114 174)(11 70 109 67)(12 169 110 172)(13 128 16 202)(14 204 17 130)(15 132 18 200)(19 159 211 162)(20 103 212 106)(21 157 213 160)(22 107 214 104)(23 161 215 158)(24 105 216 108)(25 140 28 134)(26 136 29 142)(27 144 30 138)(31 122 34 61)(32 63 35 124)(33 126 36 65)(37 56 40 190)(38 192 41 58)(39 60 42 188)(43 76 46 119)(44 115 47 78)(45 74 48 117)(49 184 52 206)(50 208 53 186)(51 182 54 210)(55 153 189 156)(57 151 191 154)(59 155 187 152)(62 164 123 167)(64 168 125 165)(66 166 121 163)(73 177 116 180)(75 175 118 178)(77 179 120 176)(79 133 82 139)(80 141 83 135)(81 137 84 143)(85 127 88 201)(86 203 89 129)(87 131 90 199)(91 197 94 146)(92 148 95 193)(93 195 96 150)(97 149 100 194)(98 196 101 145)(99 147 102 198)```

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

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

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

114 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3M 4A 4B 4C 4D 6A ··· 6CM order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 size 1 1 1 1 2 2 2 ··· 2 54 54 54 54 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 S3 D4 Dic3 D6 C3⋊D4 kernel C63.C2 C2×C33⋊5C4 C63 C3×C62 C2×C62 C32×C6 C62 C62 C3×C6 # reps 1 2 1 4 13 2 26 13 52

Matrix representation of C63.C2 in GL7(𝔽13)

 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0 5 9 0 0 0 0 0 0 0 12 0 0 0 0 0 0 10 1
,
 12 0 0 0 0 0 0 0 4 0 0 0 0 0 0 1 10 0 0 0 0 0 0 0 3 0 0 0 0 0 0 5 9 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 1 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 0 0 9 4
,
 8 0 0 0 0 0 0 0 4 11 0 0 0 0 0 2 9 0 0 0 0 0 0 0 2 5 0 0 0 0 0 2 11 0 0 0 0 0 0 0 11 10 0 0 0 0 0 6 2

`G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,12,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,5,0,0,0,0,0,0,9,0,0,0,0,0,0,0,12,10,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,10,0,0,0,0,0,0,0,3,5,0,0,0,0,0,0,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,9,0,0,0,0,0,0,4],[8,0,0,0,0,0,0,0,4,2,0,0,0,0,0,11,9,0,0,0,0,0,0,0,2,2,0,0,0,0,0,5,11,0,0,0,0,0,0,0,11,6,0,0,0,0,0,10,2] >;`

C63.C2 in GAP, Magma, Sage, TeX

`C_6^3.C_2`
`% in TeX`

`G:=Group("C6^3.C2");`
`// GroupNames label`

`G:=SmallGroup(432,511);`
`// by ID`

`G=gap.SmallGroup(432,511);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,1124,4037,14118]);`
`// Polycyclic`

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

׿
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