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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C30
 Chief series C1 — C2 — C10 — C30 — C60 — Q8×C15 — Q8×C30
 Lower central C1 — C2 — Q8×C30
 Upper central C1 — C2×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, C3, C4, C22, C5, C6, C6, C2×C4, Q8, C10, C10, C12, C2×C6, C15, C2×Q8, C20, C2×C10, C2×C12, C3×Q8, C30, C30, C2×C20, C5×Q8, C6×Q8, C60, C2×C30, Q8×C10, C2×C60, Q8×C15, Q8×C30
Quotients: C1, C2, C3, C22, C5, C6, Q8, C23, C10, C2×C6, C15, C2×Q8, C2×C10, C3×Q8, C22×C6, C30, C5×Q8, C22×C10, C6×Q8, C2×C30, Q8×C10, Q8×C15, 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)
(1 170 238 138)(2 171 239 139)(3 172 240 140)(4 173 211 141)(5 174 212 142)(6 175 213 143)(7 176 214 144)(8 177 215 145)(9 178 216 146)(10 179 217 147)(11 180 218 148)(12 151 219 149)(13 152 220 150)(14 153 221 121)(15 154 222 122)(16 155 223 123)(17 156 224 124)(18 157 225 125)(19 158 226 126)(20 159 227 127)(21 160 228 128)(22 161 229 129)(23 162 230 130)(24 163 231 131)(25 164 232 132)(26 165 233 133)(27 166 234 134)(28 167 235 135)(29 168 236 136)(30 169 237 137)(31 72 117 199)(32 73 118 200)(33 74 119 201)(34 75 120 202)(35 76 91 203)(36 77 92 204)(37 78 93 205)(38 79 94 206)(39 80 95 207)(40 81 96 208)(41 82 97 209)(42 83 98 210)(43 84 99 181)(44 85 100 182)(45 86 101 183)(46 87 102 184)(47 88 103 185)(48 89 104 186)(49 90 105 187)(50 61 106 188)(51 62 107 189)(52 63 108 190)(53 64 109 191)(54 65 110 192)(55 66 111 193)(56 67 112 194)(57 68 113 195)(58 69 114 196)(59 70 115 197)(60 71 116 198)
(1 114 238 58)(2 115 239 59)(3 116 240 60)(4 117 211 31)(5 118 212 32)(6 119 213 33)(7 120 214 34)(8 91 215 35)(9 92 216 36)(10 93 217 37)(11 94 218 38)(12 95 219 39)(13 96 220 40)(14 97 221 41)(15 98 222 42)(16 99 223 43)(17 100 224 44)(18 101 225 45)(19 102 226 46)(20 103 227 47)(21 104 228 48)(22 105 229 49)(23 106 230 50)(24 107 231 51)(25 108 232 52)(26 109 233 53)(27 110 234 54)(28 111 235 55)(29 112 236 56)(30 113 237 57)(61 130 188 162)(62 131 189 163)(63 132 190 164)(64 133 191 165)(65 134 192 166)(66 135 193 167)(67 136 194 168)(68 137 195 169)(69 138 196 170)(70 139 197 171)(71 140 198 172)(72 141 199 173)(73 142 200 174)(74 143 201 175)(75 144 202 176)(76 145 203 177)(77 146 204 178)(78 147 205 179)(79 148 206 180)(80 149 207 151)(81 150 208 152)(82 121 209 153)(83 122 210 154)(84 123 181 155)(85 124 182 156)(86 125 183 157)(87 126 184 158)(88 127 185 159)(89 128 186 160)(90 129 187 161)

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

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

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

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

150 conjugacy classes

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

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 Q8 C3×Q8 C5×Q8 Q8×C15 kernel Q8×C30 C2×C60 Q8×C15 Q8×C10 C6×Q8 C2×C20 C5×Q8 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C30 C10 C6 C2 # reps 1 3 4 2 4 6 8 12 16 8 24 32 2 4 8 16

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

 14 0 0 0 3 0 0 0 3
,
 60 0 0 0 0 1 0 60 0
,
 1 0 0 0 17 25 0 25 44
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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