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G = C30.D8order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.13D8, C60.76D4, C12.7D20, D206Dic3, C30.1SD16, C12.1(C4×D5), (C3×D20)⋊10C4, C4⋊Dic310D5, (C6×D20).6C2, (C2×D20).7S3, (C2×C20).46D6, (C2×C30).11D4, C6.4(Q8⋊D5), C4.6(D5×Dic3), C153(D4⋊C4), C33(D206C4), C60.120(C2×C4), C6.12(D4⋊D5), (C2×C12).47D10, C20.9(C3⋊D4), C2.1(C15⋊D8), C53(D4⋊Dic3), C10.12(D4⋊S3), C4.21(C3⋊D20), C10.4(D4.S3), C20.22(C2×Dic3), C30.47(C22⋊C4), (C2×C60).181C22, C2.1(C30.D4), C6.26(D10⋊C4), C2.5(D10⋊Dic3), C22.12(C15⋊D4), C10.15(C6.D4), (C5×C4⋊Dic3)⋊7C2, (C2×C153C8)⋊13C2, (C2×C4).184(S3×D5), (C2×C6).43(C5⋊D4), (C2×C10).43(C3⋊D4), SmallGroup(480,40)

Series: Derived Chief Lower central Upper central

C1C60 — C30.D8
C1C5C15C30C2×C30C2×C60C6×D20 — C30.D8
C15C30C60 — C30.D8
C1C22C2×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 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×3], C23, D5 [×2], C10 [×3], Dic3, C12 [×2], C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, D10 [×4], C2×C10, C3⋊C8, C2×Dic3, C2×C12, C3×D4 [×3], C22×C6, C3×D5 [×2], C30 [×3], D4⋊C4, C52C8, D20 [×2], D20, C2×C20, C2×C20, C22×D5, C2×C3⋊C8, C4⋊Dic3, C6×D4, C5×Dic3, C60 [×2], C6×D5 [×4], C2×C30, C2×C52C8, C5×C4⋊C4, C2×D20, D4⋊Dic3, C153C8, C3×D20 [×2], C3×D20, C10×Dic3, C2×C60, D5×C2×C6, D206C4, C5×C4⋊Dic3, C2×C153C8, C6×D20, C30.D8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D8, SD16, D10, C2×Dic3, C3⋊D4 [×2], 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)
(1 166 212 145 202 85 47 102)(2 165 213 144 203 84 48 101)(3 164 214 143 204 83 49 100)(4 163 215 142 205 82 50 99)(5 162 216 141 206 81 51 98)(6 161 217 140 207 80 52 97)(7 160 218 139 208 79 53 96)(8 159 219 138 209 78 54 95)(9 158 220 137 210 77 55 94)(10 157 221 136 181 76 56 93)(11 156 222 135 182 75 57 92)(12 155 223 134 183 74 58 91)(13 154 224 133 184 73 59 120)(14 153 225 132 185 72 60 119)(15 152 226 131 186 71 31 118)(16 151 227 130 187 70 32 117)(17 180 228 129 188 69 33 116)(18 179 229 128 189 68 34 115)(19 178 230 127 190 67 35 114)(20 177 231 126 191 66 36 113)(21 176 232 125 192 65 37 112)(22 175 233 124 193 64 38 111)(23 174 234 123 194 63 39 110)(24 173 235 122 195 62 40 109)(25 172 236 121 196 61 41 108)(26 171 237 150 197 90 42 107)(27 170 238 149 198 89 43 106)(28 169 239 148 199 88 44 105)(29 168 240 147 200 87 45 104)(30 167 211 146 201 86 46 103)
(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)(31 223)(32 212)(33 231)(34 220)(35 239)(36 228)(37 217)(38 236)(39 225)(40 214)(41 233)(42 222)(43 211)(44 230)(45 219)(46 238)(47 227)(48 216)(49 235)(50 224)(51 213)(52 232)(53 221)(54 240)(55 229)(56 218)(57 237)(58 226)(59 215)(60 234)(61 139)(62 128)(63 147)(64 136)(65 125)(66 144)(67 133)(68 122)(69 141)(70 130)(71 149)(72 138)(73 127)(74 146)(75 135)(76 124)(77 143)(78 132)(79 121)(80 140)(81 129)(82 148)(83 137)(84 126)(85 145)(86 134)(87 123)(88 142)(89 131)(90 150)(91 167)(92 156)(93 175)(94 164)(95 153)(96 172)(97 161)(98 180)(99 169)(100 158)(101 177)(102 166)(103 155)(104 174)(105 163)(106 152)(107 171)(108 160)(109 179)(110 168)(111 157)(112 176)(113 165)(114 154)(115 173)(116 162)(117 151)(118 170)(119 159)(120 178)(181 208)(182 197)(183 186)(184 205)(185 194)(187 202)(188 191)(189 210)(190 199)(192 207)(193 196)(195 204)(198 201)(200 209)(203 206)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C6D6E6F6G8A8B8C8D10A···10F12A12B15A15B20A20B20C20D20E···20L30A···30F60A···60H
order12222234444556666666888810···10121215152020202020···2030···3060···60
size1111202022212122222220202020303030302···24444444412···124···44···4

60 irreducible representations

dim11111222222222222224444444444
type++++++++-+++++-+++-+-
imageC1C2C2C2C4S3D4D4D5Dic3D6D8SD16D10C3⋊D4C3⋊D4C4×D5D20C5⋊D4D4⋊S3D4.S3S3×D5D4⋊D5Q8⋊D5D5×Dic3C3⋊D20C15⋊D4C15⋊D8C30.D4
kernelC30.D8C5×C4⋊Dic3C2×C153C8C6×D20C3×D20C2×D20C60C2×C30C4⋊Dic3D20C2×C20C30C30C2×C12C20C2×C10C12C12C2×C6C10C10C2×C4C6C6C4C4C22C2C2
# reps11114111221222224441122222244

Matrix representation of C30.D8 in GL6(𝔽241)

24000000
02400000
0019024000
001000
00000240
00001240
,
192220000
19190000
00787800
0019716300
000001
000010
,
24000000
010000
001905100
0015100
000010
000001

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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