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## G = C30.D8order 480 = 25·3·5

### 13rd non-split extension by C30 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C30.D8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C6×D20 — C30.D8
 Lower central C15 — C30 — C60 — C30.D8
 Upper central C1 — C22 — C2×C4

Generators and relations for C30.D8
G = < a,b,c | a30=b8=c2=1, bab-1=a-1, cac=a19, cbc=a15b-1 >

Subgroups: 508 in 100 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, D10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, D4⋊C4, C52C8, D20, D20, C2×C20, C2×C20, C22×D5, C2×C3⋊C8, C4⋊Dic3, C6×D4, C5×Dic3, C60, C6×D5, C2×C30, C2×C52C8, C5×C4⋊C4, C2×D20, D4⋊Dic3, C153C8, C3×D20, C3×D20, C10×Dic3, C2×C60, D5×C2×C6, D206C4, C5×C4⋊Dic3, C2×C153C8, C6×D20, C30.D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D8, SD16, D10, C2×Dic3, C3⋊D4, D4⋊C4, C4×D5, D20, C5⋊D4, D4⋊S3, D4.S3, C6.D4, S3×D5, D10⋊C4, D4⋊D5, Q8⋊D5, D4⋊Dic3, D5×Dic3, C15⋊D4, C3⋊D20, D206C4, C15⋊D8, C30.D4, D10⋊Dic3, C30.D8

Smallest permutation representation of C30.D8
On 240 points
Generators in S240
```(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 76 44 210 231 107 154 140)(2 75 45 209 232 106 155 139)(3 74 46 208 233 105 156 138)(4 73 47 207 234 104 157 137)(5 72 48 206 235 103 158 136)(6 71 49 205 236 102 159 135)(7 70 50 204 237 101 160 134)(8 69 51 203 238 100 161 133)(9 68 52 202 239 99 162 132)(10 67 53 201 240 98 163 131)(11 66 54 200 211 97 164 130)(12 65 55 199 212 96 165 129)(13 64 56 198 213 95 166 128)(14 63 57 197 214 94 167 127)(15 62 58 196 215 93 168 126)(16 61 59 195 216 92 169 125)(17 90 60 194 217 91 170 124)(18 89 31 193 218 120 171 123)(19 88 32 192 219 119 172 122)(20 87 33 191 220 118 173 121)(21 86 34 190 221 117 174 150)(22 85 35 189 222 116 175 149)(23 84 36 188 223 115 176 148)(24 83 37 187 224 114 177 147)(25 82 38 186 225 113 178 146)(26 81 39 185 226 112 179 145)(27 80 40 184 227 111 180 144)(28 79 41 183 228 110 151 143)(29 78 42 182 229 109 152 142)(30 77 43 181 230 108 153 141)
(1 16)(2 5)(3 24)(4 13)(6 21)(7 10)(8 29)(9 18)(11 26)(12 15)(14 23)(17 20)(19 28)(22 25)(27 30)(31 162)(32 151)(33 170)(34 159)(35 178)(36 167)(37 156)(38 175)(39 164)(40 153)(41 172)(42 161)(43 180)(44 169)(45 158)(46 177)(47 166)(48 155)(49 174)(50 163)(51 152)(52 171)(53 160)(54 179)(55 168)(56 157)(57 176)(58 165)(59 154)(60 173)(61 125)(62 144)(63 133)(64 122)(65 141)(66 130)(67 149)(68 138)(69 127)(70 146)(71 135)(72 124)(73 143)(74 132)(75 121)(76 140)(77 129)(78 148)(79 137)(80 126)(81 145)(82 134)(83 123)(84 142)(85 131)(86 150)(87 139)(88 128)(89 147)(90 136)(91 206)(92 195)(93 184)(94 203)(95 192)(96 181)(97 200)(98 189)(99 208)(100 197)(101 186)(102 205)(103 194)(104 183)(105 202)(106 191)(107 210)(108 199)(109 188)(110 207)(111 196)(112 185)(113 204)(114 193)(115 182)(116 201)(117 190)(118 209)(119 198)(120 187)(211 226)(212 215)(213 234)(214 223)(216 231)(217 220)(218 239)(219 228)(221 236)(222 225)(224 233)(227 230)(229 238)(232 235)(237 240)```

`G:=sub<Sym(240)| (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,76,44,210,231,107,154,140)(2,75,45,209,232,106,155,139)(3,74,46,208,233,105,156,138)(4,73,47,207,234,104,157,137)(5,72,48,206,235,103,158,136)(6,71,49,205,236,102,159,135)(7,70,50,204,237,101,160,134)(8,69,51,203,238,100,161,133)(9,68,52,202,239,99,162,132)(10,67,53,201,240,98,163,131)(11,66,54,200,211,97,164,130)(12,65,55,199,212,96,165,129)(13,64,56,198,213,95,166,128)(14,63,57,197,214,94,167,127)(15,62,58,196,215,93,168,126)(16,61,59,195,216,92,169,125)(17,90,60,194,217,91,170,124)(18,89,31,193,218,120,171,123)(19,88,32,192,219,119,172,122)(20,87,33,191,220,118,173,121)(21,86,34,190,221,117,174,150)(22,85,35,189,222,116,175,149)(23,84,36,188,223,115,176,148)(24,83,37,187,224,114,177,147)(25,82,38,186,225,113,178,146)(26,81,39,185,226,112,179,145)(27,80,40,184,227,111,180,144)(28,79,41,183,228,110,151,143)(29,78,42,182,229,109,152,142)(30,77,43,181,230,108,153,141), (1,16)(2,5)(3,24)(4,13)(6,21)(7,10)(8,29)(9,18)(11,26)(12,15)(14,23)(17,20)(19,28)(22,25)(27,30)(31,162)(32,151)(33,170)(34,159)(35,178)(36,167)(37,156)(38,175)(39,164)(40,153)(41,172)(42,161)(43,180)(44,169)(45,158)(46,177)(47,166)(48,155)(49,174)(50,163)(51,152)(52,171)(53,160)(54,179)(55,168)(56,157)(57,176)(58,165)(59,154)(60,173)(61,125)(62,144)(63,133)(64,122)(65,141)(66,130)(67,149)(68,138)(69,127)(70,146)(71,135)(72,124)(73,143)(74,132)(75,121)(76,140)(77,129)(78,148)(79,137)(80,126)(81,145)(82,134)(83,123)(84,142)(85,131)(86,150)(87,139)(88,128)(89,147)(90,136)(91,206)(92,195)(93,184)(94,203)(95,192)(96,181)(97,200)(98,189)(99,208)(100,197)(101,186)(102,205)(103,194)(104,183)(105,202)(106,191)(107,210)(108,199)(109,188)(110,207)(111,196)(112,185)(113,204)(114,193)(115,182)(116,201)(117,190)(118,209)(119,198)(120,187)(211,226)(212,215)(213,234)(214,223)(216,231)(217,220)(218,239)(219,228)(221,236)(222,225)(224,233)(227,230)(229,238)(232,235)(237,240)>;`

`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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,76,44,210,231,107,154,140)(2,75,45,209,232,106,155,139)(3,74,46,208,233,105,156,138)(4,73,47,207,234,104,157,137)(5,72,48,206,235,103,158,136)(6,71,49,205,236,102,159,135)(7,70,50,204,237,101,160,134)(8,69,51,203,238,100,161,133)(9,68,52,202,239,99,162,132)(10,67,53,201,240,98,163,131)(11,66,54,200,211,97,164,130)(12,65,55,199,212,96,165,129)(13,64,56,198,213,95,166,128)(14,63,57,197,214,94,167,127)(15,62,58,196,215,93,168,126)(16,61,59,195,216,92,169,125)(17,90,60,194,217,91,170,124)(18,89,31,193,218,120,171,123)(19,88,32,192,219,119,172,122)(20,87,33,191,220,118,173,121)(21,86,34,190,221,117,174,150)(22,85,35,189,222,116,175,149)(23,84,36,188,223,115,176,148)(24,83,37,187,224,114,177,147)(25,82,38,186,225,113,178,146)(26,81,39,185,226,112,179,145)(27,80,40,184,227,111,180,144)(28,79,41,183,228,110,151,143)(29,78,42,182,229,109,152,142)(30,77,43,181,230,108,153,141), (1,16)(2,5)(3,24)(4,13)(6,21)(7,10)(8,29)(9,18)(11,26)(12,15)(14,23)(17,20)(19,28)(22,25)(27,30)(31,162)(32,151)(33,170)(34,159)(35,178)(36,167)(37,156)(38,175)(39,164)(40,153)(41,172)(42,161)(43,180)(44,169)(45,158)(46,177)(47,166)(48,155)(49,174)(50,163)(51,152)(52,171)(53,160)(54,179)(55,168)(56,157)(57,176)(58,165)(59,154)(60,173)(61,125)(62,144)(63,133)(64,122)(65,141)(66,130)(67,149)(68,138)(69,127)(70,146)(71,135)(72,124)(73,143)(74,132)(75,121)(76,140)(77,129)(78,148)(79,137)(80,126)(81,145)(82,134)(83,123)(84,142)(85,131)(86,150)(87,139)(88,128)(89,147)(90,136)(91,206)(92,195)(93,184)(94,203)(95,192)(96,181)(97,200)(98,189)(99,208)(100,197)(101,186)(102,205)(103,194)(104,183)(105,202)(106,191)(107,210)(108,199)(109,188)(110,207)(111,196)(112,185)(113,204)(114,193)(115,182)(116,201)(117,190)(118,209)(119,198)(120,187)(211,226)(212,215)(213,234)(214,223)(216,231)(217,220)(218,239)(219,228)(221,236)(222,225)(224,233)(227,230)(229,238)(232,235)(237,240) );`

`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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,76,44,210,231,107,154,140),(2,75,45,209,232,106,155,139),(3,74,46,208,233,105,156,138),(4,73,47,207,234,104,157,137),(5,72,48,206,235,103,158,136),(6,71,49,205,236,102,159,135),(7,70,50,204,237,101,160,134),(8,69,51,203,238,100,161,133),(9,68,52,202,239,99,162,132),(10,67,53,201,240,98,163,131),(11,66,54,200,211,97,164,130),(12,65,55,199,212,96,165,129),(13,64,56,198,213,95,166,128),(14,63,57,197,214,94,167,127),(15,62,58,196,215,93,168,126),(16,61,59,195,216,92,169,125),(17,90,60,194,217,91,170,124),(18,89,31,193,218,120,171,123),(19,88,32,192,219,119,172,122),(20,87,33,191,220,118,173,121),(21,86,34,190,221,117,174,150),(22,85,35,189,222,116,175,149),(23,84,36,188,223,115,176,148),(24,83,37,187,224,114,177,147),(25,82,38,186,225,113,178,146),(26,81,39,185,226,112,179,145),(27,80,40,184,227,111,180,144),(28,79,41,183,228,110,151,143),(29,78,42,182,229,109,152,142),(30,77,43,181,230,108,153,141)], [(1,16),(2,5),(3,24),(4,13),(6,21),(7,10),(8,29),(9,18),(11,26),(12,15),(14,23),(17,20),(19,28),(22,25),(27,30),(31,162),(32,151),(33,170),(34,159),(35,178),(36,167),(37,156),(38,175),(39,164),(40,153),(41,172),(42,161),(43,180),(44,169),(45,158),(46,177),(47,166),(48,155),(49,174),(50,163),(51,152),(52,171),(53,160),(54,179),(55,168),(56,157),(57,176),(58,165),(59,154),(60,173),(61,125),(62,144),(63,133),(64,122),(65,141),(66,130),(67,149),(68,138),(69,127),(70,146),(71,135),(72,124),(73,143),(74,132),(75,121),(76,140),(77,129),(78,148),(79,137),(80,126),(81,145),(82,134),(83,123),(84,142),(85,131),(86,150),(87,139),(88,128),(89,147),(90,136),(91,206),(92,195),(93,184),(94,203),(95,192),(96,181),(97,200),(98,189),(99,208),(100,197),(101,186),(102,205),(103,194),(104,183),(105,202),(106,191),(107,210),(108,199),(109,188),(110,207),(111,196),(112,185),(113,204),(114,193),(115,182),(116,201),(117,190),(118,209),(119,198),(120,187),(211,226),(212,215),(213,234),(214,223),(216,231),(217,220),(218,239),(219,228),(221,236),(222,225),(224,233),(227,230),(229,238),(232,235),(237,240)]])`

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A ··· 10F 12A 12B 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 20 20 2 2 2 12 12 2 2 2 2 2 20 20 20 20 30 30 30 30 2 ··· 2 4 4 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + - + + + - + - image C1 C2 C2 C2 C4 S3 D4 D4 D5 Dic3 D6 D8 SD16 D10 C3⋊D4 C3⋊D4 C4×D5 D20 C5⋊D4 D4⋊S3 D4.S3 S3×D5 D4⋊D5 Q8⋊D5 D5×Dic3 C3⋊D20 C15⋊D4 C15⋊D8 C30.D4 kernel C30.D8 C5×C4⋊Dic3 C2×C15⋊3C8 C6×D20 C3×D20 C2×D20 C60 C2×C30 C4⋊Dic3 D20 C2×C20 C30 C30 C2×C12 C20 C2×C10 C12 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 2 1 2 2 2 2 2 4 4 4 1 1 2 2 2 2 2 2 4 4

Matrix representation of C30.D8 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 190 240 0 0 0 0 1 0 0 0 0 0 0 0 0 240 0 0 0 0 1 240
,
 19 222 0 0 0 0 19 19 0 0 0 0 0 0 78 78 0 0 0 0 197 163 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 240 0 0 0 0 0 0 1 0 0 0 0 0 0 190 51 0 0 0 0 1 51 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,190,1,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[19,19,0,0,0,0,222,19,0,0,0,0,0,0,78,197,0,0,0,0,78,163,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[240,0,0,0,0,0,0,1,0,0,0,0,0,0,190,1,0,0,0,0,51,51,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C30.D8 in GAP, Magma, Sage, TeX

`C_{30}.D_8`
`% in TeX`

`G:=Group("C30.D8");`
`// GroupNames label`

`G:=SmallGroup(480,40);`
`// by ID`

`G=gap.SmallGroup(480,40);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,675,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^30=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^19,c*b*c=a^15*b^-1>;`
`// generators/relations`

׿
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