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## G = C15×C4⋊C4order 240 = 24·3·5

### Direct product of C15 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C15×C4⋊C4
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C2×C60 — C15×C4⋊C4
 Lower central C1 — C2 — C15×C4⋊C4
 Upper central C1 — C2×C30 — C15×C4⋊C4

Generators and relations for C15×C4⋊C4
G = < a,b,c | a15=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

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

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

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

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

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 2 2 2 2 type + + + - image C1 C2 C3 C4 C5 C6 C10 C12 C15 C20 C30 C60 D4 Q8 C3×D4 C3×Q8 C5×D4 C5×Q8 D4×C15 Q8×C15 kernel C15×C4⋊C4 C2×C60 C5×C4⋊C4 C60 C3×C4⋊C4 C2×C20 C2×C12 C20 C4⋊C4 C12 C2×C4 C4 C30 C30 C10 C10 C6 C6 C2 C2 # reps 1 3 2 4 4 6 12 8 8 16 24 32 1 1 2 2 4 4 8 8

Matrix representation of C15×C4⋊C4 in GL3(𝔽61) generated by

 1 0 0 0 25 0 0 0 25
,
 60 0 0 0 1 2 0 60 60
,
 50 0 0 0 47 16 0 22 14
G:=sub<GL(3,GF(61))| [1,0,0,0,25,0,0,0,25],[60,0,0,0,1,60,0,2,60],[50,0,0,0,47,22,0,16,14] >;

C15×C4⋊C4 in GAP, Magma, Sage, TeX

C_{15}\times C_4\rtimes C_4
% in TeX

G:=Group("C15xC4:C4");
// GroupNames label

G:=SmallGroup(240,83);
// by ID

G=gap.SmallGroup(240,83);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-2,720,745,367]);
// Polycyclic

G:=Group<a,b,c|a^15=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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