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G = D303C8order 480 = 25·3·5

1st semidirect product of D30 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D303C8, C4.19D60, C12.37D20, C20.37D12, C60.209D4, C30.27M4(2), (C2×C40)⋊3S3, C54(D6⋊C8), (C2×C24)⋊3D5, (C2×C8)⋊1D15, (C2×C120)⋊5C2, C2.5(C8×D15), C6.10(C8×D5), C10.19(S3×C8), C30.47(C2×C8), (C2×C4).94D30, C32(D101C8), C1510(C22⋊C8), (C2×C20).408D6, C6.6(C8⋊D5), C2.3(C40⋊S3), C10.32(D6⋊C4), (C2×C12).412D10, C4.27(C157D4), (C22×D15).8C4, C22.11(C4×D15), C10.11(C8⋊S3), C2.1(D303C4), C20.106(C3⋊D4), C12.106(C5⋊D4), C30.74(C22⋊C4), (C2×C60).494C22, (C2×Dic15).14C4, C6.17(D10⋊C4), (C2×C4×D15).6C2, (C2×C153C8)⋊9C2, (C2×C6).29(C4×D5), (C2×C10).54(C4×S3), (C2×C30).136(C2×C4), SmallGroup(480,180)

Series: Derived Chief Lower central Upper central

C1C30 — D303C8
C1C5C15C30C60C2×C60C2×C4×D15 — D303C8
C15C30 — D303C8
C1C2×C4C2×C8

Generators and relations for D303C8
 G = < a,b,c | a30=b2=c8=1, bab=a-1, ac=ca, cbc-1=a15b >

Subgroups: 596 in 100 conjugacy classes, 43 normal (41 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4 [×3], C23, D5 [×2], C10 [×3], Dic3, C12 [×2], D6 [×4], C2×C6, C15, C2×C8, C2×C8, C22×C4, Dic5, C20 [×2], D10 [×4], C2×C10, C3⋊C8, C24, C4×S3 [×2], C2×Dic3, C2×C12, C22×S3, D15 [×2], C30 [×3], C22⋊C8, C52C8, C40, C4×D5 [×2], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15, C60 [×2], D30 [×2], D30 [×2], C2×C30, C2×C52C8, C2×C40, C2×C4×D5, D6⋊C8, C153C8, C120, C4×D15 [×2], C2×Dic15, C2×C60, C22×D15, D101C8, C2×C153C8, C2×C120, C2×C4×D15, D303C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D5, D6, C22⋊C4, C2×C8, M4(2), D10, C4×S3, D12, C3⋊D4, D15, C22⋊C8, C4×D5, D20, C5⋊D4, S3×C8, C8⋊S3, D6⋊C4, D30, C8×D5, C8⋊D5, D10⋊C4, D6⋊C8, C4×D15, D60, C157D4, D101C8, C8×D15, C40⋊S3, D303C4, D303C8

Smallest permutation representation of D303C8
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)
(1 180)(2 179)(3 178)(4 177)(5 176)(6 175)(7 174)(8 173)(9 172)(10 171)(11 170)(12 169)(13 168)(14 167)(15 166)(16 165)(17 164)(18 163)(19 162)(20 161)(21 160)(22 159)(23 158)(24 157)(25 156)(26 155)(27 154)(28 153)(29 152)(30 151)(31 223)(32 222)(33 221)(34 220)(35 219)(36 218)(37 217)(38 216)(39 215)(40 214)(41 213)(42 212)(43 211)(44 240)(45 239)(46 238)(47 237)(48 236)(49 235)(50 234)(51 233)(52 232)(53 231)(54 230)(55 229)(56 228)(57 227)(58 226)(59 225)(60 224)(61 92)(62 91)(63 120)(64 119)(65 118)(66 117)(67 116)(68 115)(69 114)(70 113)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(121 208)(122 207)(123 206)(124 205)(125 204)(126 203)(127 202)(128 201)(129 200)(130 199)(131 198)(132 197)(133 196)(134 195)(135 194)(136 193)(137 192)(138 191)(139 190)(140 189)(141 188)(142 187)(143 186)(144 185)(145 184)(146 183)(147 182)(148 181)(149 210)(150 209)
(1 81 46 134 151 118 239 181)(2 82 47 135 152 119 240 182)(3 83 48 136 153 120 211 183)(4 84 49 137 154 91 212 184)(5 85 50 138 155 92 213 185)(6 86 51 139 156 93 214 186)(7 87 52 140 157 94 215 187)(8 88 53 141 158 95 216 188)(9 89 54 142 159 96 217 189)(10 90 55 143 160 97 218 190)(11 61 56 144 161 98 219 191)(12 62 57 145 162 99 220 192)(13 63 58 146 163 100 221 193)(14 64 59 147 164 101 222 194)(15 65 60 148 165 102 223 195)(16 66 31 149 166 103 224 196)(17 67 32 150 167 104 225 197)(18 68 33 121 168 105 226 198)(19 69 34 122 169 106 227 199)(20 70 35 123 170 107 228 200)(21 71 36 124 171 108 229 201)(22 72 37 125 172 109 230 202)(23 73 38 126 173 110 231 203)(24 74 39 127 174 111 232 204)(25 75 40 128 175 112 233 205)(26 76 41 129 176 113 234 206)(27 77 42 130 177 114 235 207)(28 78 43 131 178 115 236 208)(29 79 44 132 179 116 237 209)(30 80 45 133 180 117 238 210)

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

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

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

132 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D8E8F8G8H10A···10F12A12B12C12D15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order1222223444444556668888888810···10121212121515151520···2024···2430···3040···4060···60120···120
size11113030211113030222222222303030302···2222222222···22···22···22···22···22···2

132 irreducible representations

dim111111122222222222222222222222
type++++++++++++++
imageC1C2C2C2C4C4C8S3D4D5D6M4(2)D10D12C3⋊D4C4×S3D15D20C5⋊D4C4×D5S3×C8C8⋊S3D30C8×D5C8⋊D5D60C157D4C4×D15C8×D15C40⋊S3
kernelD303C8C2×C153C8C2×C120C2×C4×D15C2×Dic15C22×D15D30C2×C40C60C2×C24C2×C20C30C2×C12C20C20C2×C10C2×C8C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11112281221222224444444888881616

Matrix representation of D303C8 in GL3(𝔽241) generated by

100
017363
017830
,
100
017363
022568
,
21100
015443
019887
G:=sub<GL(3,GF(241))| [1,0,0,0,173,178,0,63,30],[1,0,0,0,173,225,0,63,68],[211,0,0,0,154,198,0,43,87] >;

D303C8 in GAP, Magma, Sage, TeX

D_{30}\rtimes_3C_8
% in TeX

G:=Group("D30:3C8");
// GroupNames label

G:=SmallGroup(480,180);
// by ID

G=gap.SmallGroup(480,180);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,2693,18822]);
// Polycyclic

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

׿
×
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