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G = Q8×C30order 240 = 24·3·5

Direct product of C30 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C30, C30.59C23, C60.80C22, C4.4(C2×C30), (C2×C20).9C6, (C2×C4).3C30, (C2×C60).21C2, (C2×C12).9C10, C20.20(C2×C6), C12.20(C2×C10), C22.4(C2×C30), C2.2(C22×C30), (C2×C30).54C22, C10.12(C22×C6), C6.12(C22×C10), (C2×C10).15(C2×C6), (C2×C6).15(C2×C10), SmallGroup(240,187)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C30
C1C2C10C30C60Q8×C15 — Q8×C30
C1C2 — Q8×C30
C1C2×C30 — Q8×C30

Generators and relations for Q8×C30
 G = < a,b,c | a30=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76, all normal (16 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C22, C5, C6, C6 [×2], C2×C4 [×3], Q8 [×4], C10, C10 [×2], C12 [×6], C2×C6, C15, C2×Q8, C20 [×6], C2×C10, C2×C12 [×3], C3×Q8 [×4], C30, C30 [×2], C2×C20 [×3], C5×Q8 [×4], C6×Q8, C60 [×6], C2×C30, Q8×C10, C2×C60 [×3], Q8×C15 [×4], Q8×C30
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], Q8 [×2], C23, C10 [×7], C2×C6 [×7], C15, C2×Q8, C2×C10 [×7], C3×Q8 [×2], C22×C6, C30 [×7], C5×Q8 [×2], C22×C10, C6×Q8, C2×C30 [×7], Q8×C10, Q8×C15 [×2], C22×C30, Q8×C30

Smallest permutation representation of Q8×C30
Regular action 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)
(1 120 73 52)(2 91 74 53)(3 92 75 54)(4 93 76 55)(5 94 77 56)(6 95 78 57)(7 96 79 58)(8 97 80 59)(9 98 81 60)(10 99 82 31)(11 100 83 32)(12 101 84 33)(13 102 85 34)(14 103 86 35)(15 104 87 36)(16 105 88 37)(17 106 89 38)(18 107 90 39)(19 108 61 40)(20 109 62 41)(21 110 63 42)(22 111 64 43)(23 112 65 44)(24 113 66 45)(25 114 67 46)(26 115 68 47)(27 116 69 48)(28 117 70 49)(29 118 71 50)(30 119 72 51)(121 211 157 209)(122 212 158 210)(123 213 159 181)(124 214 160 182)(125 215 161 183)(126 216 162 184)(127 217 163 185)(128 218 164 186)(129 219 165 187)(130 220 166 188)(131 221 167 189)(132 222 168 190)(133 223 169 191)(134 224 170 192)(135 225 171 193)(136 226 172 194)(137 227 173 195)(138 228 174 196)(139 229 175 197)(140 230 176 198)(141 231 177 199)(142 232 178 200)(143 233 179 201)(144 234 180 202)(145 235 151 203)(146 236 152 204)(147 237 153 205)(148 238 154 206)(149 239 155 207)(150 240 156 208)
(1 131 73 167)(2 132 74 168)(3 133 75 169)(4 134 76 170)(5 135 77 171)(6 136 78 172)(7 137 79 173)(8 138 80 174)(9 139 81 175)(10 140 82 176)(11 141 83 177)(12 142 84 178)(13 143 85 179)(14 144 86 180)(15 145 87 151)(16 146 88 152)(17 147 89 153)(18 148 90 154)(19 149 61 155)(20 150 62 156)(21 121 63 157)(22 122 64 158)(23 123 65 159)(24 124 66 160)(25 125 67 161)(26 126 68 162)(27 127 69 163)(28 128 70 164)(29 129 71 165)(30 130 72 166)(31 230 99 198)(32 231 100 199)(33 232 101 200)(34 233 102 201)(35 234 103 202)(36 235 104 203)(37 236 105 204)(38 237 106 205)(39 238 107 206)(40 239 108 207)(41 240 109 208)(42 211 110 209)(43 212 111 210)(44 213 112 181)(45 214 113 182)(46 215 114 183)(47 216 115 184)(48 217 116 185)(49 218 117 186)(50 219 118 187)(51 220 119 188)(52 221 120 189)(53 222 91 190)(54 223 92 191)(55 224 93 192)(56 225 94 193)(57 226 95 194)(58 227 96 195)(59 228 97 196)(60 229 98 197)

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

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

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

Q8×C30 is a maximal subgroup of   Q82Dic15  C60.10D4  Q8.11D30  Dic154Q8  D307Q8  C60.23D4  Q8.15D30

150 conjugacy classes

class 1 2A2B2C3A3B4A···4F5A5B5C5D6A···6F10A···10L12A···12L15A···15H20A···20X30A···30X60A···60AV
order1222334···455556···610···1012···1215···1520···2030···3060···60
size1111112···211111···11···12···21···12···21···12···2

150 irreducible representations

dim1111111111112222
type+++-
imageC1C2C2C3C5C6C6C10C10C15C30C30Q8C3×Q8C5×Q8Q8×C15
kernelQ8×C30C2×C60Q8×C15Q8×C10C6×Q8C2×C20C5×Q8C2×C12C3×Q8C2×Q8C2×C4Q8C30C10C6C2
# reps134246812168243224816

Matrix representation of Q8×C30 in GL3(𝔽61) generated by

1400
030
003
,
6000
001
0600
,
100
01725
02544
G:=sub<GL(3,GF(61))| [14,0,0,0,3,0,0,0,3],[60,0,0,0,0,60,0,1,0],[1,0,0,0,17,25,0,25,44] >;

Q8×C30 in GAP, Magma, Sage, TeX

Q_8\times C_{30}
% in TeX

G:=Group("Q8xC30");
// GroupNames label

G:=SmallGroup(240,187);
// by ID

G=gap.SmallGroup(240,187);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-2,720,1465,727]);
// Polycyclic

G:=Group<a,b,c|a^30=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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×
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