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## G = M4(2)×C30order 480 = 25·3·5

### Direct product of C30 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C30
 Chief series C1 — C2 — C4 — C20 — C60 — C120 — C15×M4(2) — M4(2)×C30
 Lower central C1 — C2 — M4(2)×C30
 Upper central C1 — C2×C60 — M4(2)×C30

Generators and relations for M4(2)×C30
G = < a,b,c | a30=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 152 in 136 conjugacy classes, 120 normal (40 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C24, C2×C12, C2×C12, C22×C6, C30, C30, C30, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C2×C24, C3×M4(2), C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C2×C40, C5×M4(2), C22×C20, C6×M4(2), C120, C2×C60, C2×C60, C22×C30, C10×M4(2), C2×C120, C15×M4(2), C22×C60, M4(2)×C30
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C23, C10, C12, C2×C6, C15, M4(2), C22×C4, C20, C2×C10, C2×C12, C22×C6, C30, C2×M4(2), C2×C20, C22×C10, C3×M4(2), C22×C12, C60, C2×C30, C5×M4(2), C22×C20, C6×M4(2), C2×C60, C22×C30, C10×M4(2), C15×M4(2), C22×C60, M4(2)×C30

Smallest permutation representation of M4(2)×C30
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)
(1 185 144 39 100 218 79 165)(2 186 145 40 101 219 80 166)(3 187 146 41 102 220 81 167)(4 188 147 42 103 221 82 168)(5 189 148 43 104 222 83 169)(6 190 149 44 105 223 84 170)(7 191 150 45 106 224 85 171)(8 192 121 46 107 225 86 172)(9 193 122 47 108 226 87 173)(10 194 123 48 109 227 88 174)(11 195 124 49 110 228 89 175)(12 196 125 50 111 229 90 176)(13 197 126 51 112 230 61 177)(14 198 127 52 113 231 62 178)(15 199 128 53 114 232 63 179)(16 200 129 54 115 233 64 180)(17 201 130 55 116 234 65 151)(18 202 131 56 117 235 66 152)(19 203 132 57 118 236 67 153)(20 204 133 58 119 237 68 154)(21 205 134 59 120 238 69 155)(22 206 135 60 91 239 70 156)(23 207 136 31 92 240 71 157)(24 208 137 32 93 211 72 158)(25 209 138 33 94 212 73 159)(26 210 139 34 95 213 74 160)(27 181 140 35 96 214 75 161)(28 182 141 36 97 215 76 162)(29 183 142 37 98 216 77 163)(30 184 143 38 99 217 78 164)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(151 166)(152 167)(153 168)(154 169)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)(181 196)(182 197)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)(211 226)(212 227)(213 228)(214 229)(215 230)(216 231)(217 232)(218 233)(219 234)(220 235)(221 236)(222 237)(223 238)(224 239)(225 240)

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

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

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

300 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A ··· 6F 6G 6H 6I 6J 8A ··· 8H 10A ··· 10L 10M ··· 10T 12A ··· 12H 12I 12J 12K 12L 15A ··· 15H 20A ··· 20P 20Q ··· 20X 24A ··· 24P 30A ··· 30X 30Y ··· 30AN 40A ··· 40AF 60A ··· 60AF 60AG ··· 60AV 120A ··· 120BL order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 5 5 6 ··· 6 6 6 6 6 8 ··· 8 10 ··· 10 10 ··· 10 12 ··· 12 12 12 12 12 15 ··· 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 1 1 2 2 1 1 1 1 1 1 2 2 1 1 1 1 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

300 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C2 C3 C4 C4 C5 C6 C6 C6 C10 C10 C10 C12 C12 C15 C20 C20 C30 C30 C30 C60 C60 M4(2) C3×M4(2) C5×M4(2) C15×M4(2) kernel M4(2)×C30 C2×C120 C15×M4(2) C22×C60 C10×M4(2) C2×C60 C22×C30 C6×M4(2) C2×C40 C5×M4(2) C22×C20 C2×C24 C3×M4(2) C22×C12 C2×C20 C22×C10 C2×M4(2) C2×C12 C22×C6 C2×C8 M4(2) C22×C4 C2×C4 C23 C30 C10 C6 C2 # reps 1 2 4 1 2 6 2 4 4 8 2 8 16 4 12 4 8 24 8 16 32 8 48 16 4 8 16 32

Matrix representation of M4(2)×C30 in GL3(𝔽241) generated by

 240 0 0 0 231 0 0 0 231
,
 240 0 0 0 240 239 0 153 1
,
 1 0 0 0 240 0 0 1 1
G:=sub<GL(3,GF(241))| [240,0,0,0,231,0,0,0,231],[240,0,0,0,240,153,0,239,1],[1,0,0,0,240,1,0,0,1] >;

M4(2)×C30 in GAP, Magma, Sage, TeX

M_4(2)\times C_{30}
% in TeX

G:=Group("M4(2)xC30");
// GroupNames label

G:=SmallGroup(480,935);
// by ID

G=gap.SmallGroup(480,935);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,840,3389,124]);
// Polycyclic

G:=Group<a,b,c|a^30=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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