direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C4⋊C4, C4⋊C60, C60⋊11C4, C20⋊5C12, C12⋊3C20, C30.53D4, C30.10Q8, C2.(Q8×C15), C6.3(C5×Q8), C2.2(C2×C60), (C2×C20).2C6, (C2×C60).4C2, (C2×C4).1C30, C2.2(D4×C15), C6.13(C5×D4), C10.3(C3×Q8), C6.11(C2×C20), C30.63(C2×C4), (C2×C12).2C10, C10.13(C3×D4), C10.18(C2×C12), C22.3(C2×C30), (C2×C30).53C22, (C2×C6).14(C2×C10), (C2×C10).14(C2×C6), SmallGroup(240,83)
Series: Derived ►Chief ►Lower central ►Upper central
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 68 166 203)(2 69 167 204)(3 70 168 205)(4 71 169 206)(5 72 170 207)(6 73 171 208)(7 74 172 209)(8 75 173 210)(9 61 174 196)(10 62 175 197)(11 63 176 198)(12 64 177 199)(13 65 178 200)(14 66 179 201)(15 67 180 202)(16 56 156 93)(17 57 157 94)(18 58 158 95)(19 59 159 96)(20 60 160 97)(21 46 161 98)(22 47 162 99)(23 48 163 100)(24 49 164 101)(25 50 165 102)(26 51 151 103)(27 52 152 104)(28 53 153 105)(29 54 154 91)(30 55 155 92)(31 220 142 86)(32 221 143 87)(33 222 144 88)(34 223 145 89)(35 224 146 90)(36 225 147 76)(37 211 148 77)(38 212 149 78)(39 213 150 79)(40 214 136 80)(41 215 137 81)(42 216 138 82)(43 217 139 83)(44 218 140 84)(45 219 141 85)(106 182 135 237)(107 183 121 238)(108 184 122 239)(109 185 123 240)(110 186 124 226)(111 187 125 227)(112 188 126 228)(113 189 127 229)(114 190 128 230)(115 191 129 231)(116 192 130 232)(117 193 131 233)(118 194 132 234)(119 195 133 235)(120 181 134 236)
(1 221 189 21)(2 222 190 22)(3 223 191 23)(4 224 192 24)(5 225 193 25)(6 211 194 26)(7 212 195 27)(8 213 181 28)(9 214 182 29)(10 215 183 30)(11 216 184 16)(12 217 185 17)(13 218 186 18)(14 219 187 19)(15 220 188 20)(31 126 97 67)(32 127 98 68)(33 128 99 69)(34 129 100 70)(35 130 101 71)(36 131 102 72)(37 132 103 73)(38 133 104 74)(39 134 105 75)(40 135 91 61)(41 121 92 62)(42 122 93 63)(43 123 94 64)(44 124 95 65)(45 125 96 66)(46 203 143 113)(47 204 144 114)(48 205 145 115)(49 206 146 116)(50 207 147 117)(51 208 148 118)(52 209 149 119)(53 210 150 120)(54 196 136 106)(55 197 137 107)(56 198 138 108)(57 199 139 109)(58 200 140 110)(59 201 141 111)(60 202 142 112)(76 233 165 170)(77 234 151 171)(78 235 152 172)(79 236 153 173)(80 237 154 174)(81 238 155 175)(82 239 156 176)(83 240 157 177)(84 226 158 178)(85 227 159 179)(86 228 160 180)(87 229 161 166)(88 230 162 167)(89 231 163 168)(90 232 164 169)
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), (1,68,166,203)(2,69,167,204)(3,70,168,205)(4,71,169,206)(5,72,170,207)(6,73,171,208)(7,74,172,209)(8,75,173,210)(9,61,174,196)(10,62,175,197)(11,63,176,198)(12,64,177,199)(13,65,178,200)(14,66,179,201)(15,67,180,202)(16,56,156,93)(17,57,157,94)(18,58,158,95)(19,59,159,96)(20,60,160,97)(21,46,161,98)(22,47,162,99)(23,48,163,100)(24,49,164,101)(25,50,165,102)(26,51,151,103)(27,52,152,104)(28,53,153,105)(29,54,154,91)(30,55,155,92)(31,220,142,86)(32,221,143,87)(33,222,144,88)(34,223,145,89)(35,224,146,90)(36,225,147,76)(37,211,148,77)(38,212,149,78)(39,213,150,79)(40,214,136,80)(41,215,137,81)(42,216,138,82)(43,217,139,83)(44,218,140,84)(45,219,141,85)(106,182,135,237)(107,183,121,238)(108,184,122,239)(109,185,123,240)(110,186,124,226)(111,187,125,227)(112,188,126,228)(113,189,127,229)(114,190,128,230)(115,191,129,231)(116,192,130,232)(117,193,131,233)(118,194,132,234)(119,195,133,235)(120,181,134,236), (1,221,189,21)(2,222,190,22)(3,223,191,23)(4,224,192,24)(5,225,193,25)(6,211,194,26)(7,212,195,27)(8,213,181,28)(9,214,182,29)(10,215,183,30)(11,216,184,16)(12,217,185,17)(13,218,186,18)(14,219,187,19)(15,220,188,20)(31,126,97,67)(32,127,98,68)(33,128,99,69)(34,129,100,70)(35,130,101,71)(36,131,102,72)(37,132,103,73)(38,133,104,74)(39,134,105,75)(40,135,91,61)(41,121,92,62)(42,122,93,63)(43,123,94,64)(44,124,95,65)(45,125,96,66)(46,203,143,113)(47,204,144,114)(48,205,145,115)(49,206,146,116)(50,207,147,117)(51,208,148,118)(52,209,149,119)(53,210,150,120)(54,196,136,106)(55,197,137,107)(56,198,138,108)(57,199,139,109)(58,200,140,110)(59,201,141,111)(60,202,142,112)(76,233,165,170)(77,234,151,171)(78,235,152,172)(79,236,153,173)(80,237,154,174)(81,238,155,175)(82,239,156,176)(83,240,157,177)(84,226,158,178)(85,227,159,179)(86,228,160,180)(87,229,161,166)(88,230,162,167)(89,231,163,168)(90,232,164,169)>;
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), (1,68,166,203)(2,69,167,204)(3,70,168,205)(4,71,169,206)(5,72,170,207)(6,73,171,208)(7,74,172,209)(8,75,173,210)(9,61,174,196)(10,62,175,197)(11,63,176,198)(12,64,177,199)(13,65,178,200)(14,66,179,201)(15,67,180,202)(16,56,156,93)(17,57,157,94)(18,58,158,95)(19,59,159,96)(20,60,160,97)(21,46,161,98)(22,47,162,99)(23,48,163,100)(24,49,164,101)(25,50,165,102)(26,51,151,103)(27,52,152,104)(28,53,153,105)(29,54,154,91)(30,55,155,92)(31,220,142,86)(32,221,143,87)(33,222,144,88)(34,223,145,89)(35,224,146,90)(36,225,147,76)(37,211,148,77)(38,212,149,78)(39,213,150,79)(40,214,136,80)(41,215,137,81)(42,216,138,82)(43,217,139,83)(44,218,140,84)(45,219,141,85)(106,182,135,237)(107,183,121,238)(108,184,122,239)(109,185,123,240)(110,186,124,226)(111,187,125,227)(112,188,126,228)(113,189,127,229)(114,190,128,230)(115,191,129,231)(116,192,130,232)(117,193,131,233)(118,194,132,234)(119,195,133,235)(120,181,134,236), (1,221,189,21)(2,222,190,22)(3,223,191,23)(4,224,192,24)(5,225,193,25)(6,211,194,26)(7,212,195,27)(8,213,181,28)(9,214,182,29)(10,215,183,30)(11,216,184,16)(12,217,185,17)(13,218,186,18)(14,219,187,19)(15,220,188,20)(31,126,97,67)(32,127,98,68)(33,128,99,69)(34,129,100,70)(35,130,101,71)(36,131,102,72)(37,132,103,73)(38,133,104,74)(39,134,105,75)(40,135,91,61)(41,121,92,62)(42,122,93,63)(43,123,94,64)(44,124,95,65)(45,125,96,66)(46,203,143,113)(47,204,144,114)(48,205,145,115)(49,206,146,116)(50,207,147,117)(51,208,148,118)(52,209,149,119)(53,210,150,120)(54,196,136,106)(55,197,137,107)(56,198,138,108)(57,199,139,109)(58,200,140,110)(59,201,141,111)(60,202,142,112)(76,233,165,170)(77,234,151,171)(78,235,152,172)(79,236,153,173)(80,237,154,174)(81,238,155,175)(82,239,156,176)(83,240,157,177)(84,226,158,178)(85,227,159,179)(86,228,160,180)(87,229,161,166)(88,230,162,167)(89,231,163,168)(90,232,164,169) );
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)], [(1,68,166,203),(2,69,167,204),(3,70,168,205),(4,71,169,206),(5,72,170,207),(6,73,171,208),(7,74,172,209),(8,75,173,210),(9,61,174,196),(10,62,175,197),(11,63,176,198),(12,64,177,199),(13,65,178,200),(14,66,179,201),(15,67,180,202),(16,56,156,93),(17,57,157,94),(18,58,158,95),(19,59,159,96),(20,60,160,97),(21,46,161,98),(22,47,162,99),(23,48,163,100),(24,49,164,101),(25,50,165,102),(26,51,151,103),(27,52,152,104),(28,53,153,105),(29,54,154,91),(30,55,155,92),(31,220,142,86),(32,221,143,87),(33,222,144,88),(34,223,145,89),(35,224,146,90),(36,225,147,76),(37,211,148,77),(38,212,149,78),(39,213,150,79),(40,214,136,80),(41,215,137,81),(42,216,138,82),(43,217,139,83),(44,218,140,84),(45,219,141,85),(106,182,135,237),(107,183,121,238),(108,184,122,239),(109,185,123,240),(110,186,124,226),(111,187,125,227),(112,188,126,228),(113,189,127,229),(114,190,128,230),(115,191,129,231),(116,192,130,232),(117,193,131,233),(118,194,132,234),(119,195,133,235),(120,181,134,236)], [(1,221,189,21),(2,222,190,22),(3,223,191,23),(4,224,192,24),(5,225,193,25),(6,211,194,26),(7,212,195,27),(8,213,181,28),(9,214,182,29),(10,215,183,30),(11,216,184,16),(12,217,185,17),(13,218,186,18),(14,219,187,19),(15,220,188,20),(31,126,97,67),(32,127,98,68),(33,128,99,69),(34,129,100,70),(35,130,101,71),(36,131,102,72),(37,132,103,73),(38,133,104,74),(39,134,105,75),(40,135,91,61),(41,121,92,62),(42,122,93,63),(43,123,94,64),(44,124,95,65),(45,125,96,66),(46,203,143,113),(47,204,144,114),(48,205,145,115),(49,206,146,116),(50,207,147,117),(51,208,148,118),(52,209,149,119),(53,210,150,120),(54,196,136,106),(55,197,137,107),(56,198,138,108),(57,199,139,109),(58,200,140,110),(59,201,141,111),(60,202,142,112),(76,233,165,170),(77,234,151,171),(78,235,152,172),(79,236,153,173),(80,237,154,174),(81,238,155,175),(82,239,156,176),(83,240,157,177),(84,226,158,178),(85,227,159,179),(86,228,160,180),(87,229,161,166),(88,230,162,167),(89,231,163,168),(90,232,164,169)]])
C15×C4⋊C4 is a maximal subgroup of
C60.1Q8 C60.2Q8 D60⋊9C4 Dic30⋊9C4 Dic15⋊10Q8 C4⋊Dic30 Dic15.3Q8 C4.Dic30 C4⋊C4⋊7D15 D60⋊11C4 D30.29D4 C4⋊D60 D30⋊5Q8 D30⋊6Q8 C4⋊C4⋊D15 D4×C60 Q8×C60
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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