direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×C3⋊Dic3, C30.7S3, C32⋊3C20, C15⋊5Dic3, C3⋊(C5×Dic3), (C3×C15)⋊11C4, C6.3(C5×S3), (C3×C30).5C2, (C3×C6).2C10, C10.2(C3⋊S3), C2.(C5×C3⋊S3), SmallGroup(180,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C5×C3⋊Dic3 |
Generators and relations for C5×C3⋊Dic3
G = < a,b,c,d | a5=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Smallest permutation representation of C5×C3⋊Dic3
►Regular action on 180 points
Generators in S180
(1 98 80 62 44)(2 99 81 63 45)(3 100 82 64 46)(4 101 83 65 47)(5 102 84 66 48)(6 97 79 61 43)(7 174 156 138 120)(8 169 151 133 115)(9 170 152 134 116)(10 171 153 135 117)(11 172 154 136 118)(12 173 155 137 119)(13 168 150 132 114)(14 163 145 127 109)(15 164 146 128 110)(16 165 147 129 111)(17 166 148 130 112)(18 167 149 131 113)(19 94 73 55 37)(20 95 74 56 38)(21 96 75 57 39)(22 91 76 58 40)(23 92 77 59 41)(24 93 78 60 42)(25 89 71 53 35)(26 90 72 54 36)(27 85 67 49 31)(28 86 68 50 32)(29 87 69 51 33)(30 88 70 52 34)(103 175 157 139 121)(104 176 158 140 122)(105 177 159 141 123)(106 178 160 142 124)(107 179 161 143 125)(108 180 162 144 126)
(1 23 29)(2 24 30)(3 19 25)(4 20 26)(5 21 27)(6 22 28)(7 175 14)(8 176 15)(9 177 16)(10 178 17)(11 179 18)(12 180 13)(31 48 39)(32 43 40)(33 44 41)(34 45 42)(35 46 37)(36 47 38)(49 66 57)(50 61 58)(51 62 59)(52 63 60)(53 64 55)(54 65 56)(67 84 75)(68 79 76)(69 80 77)(70 81 78)(71 82 73)(72 83 74)(85 102 96)(86 97 91)(87 98 92)(88 99 93)(89 100 94)(90 101 95)(103 109 120)(104 110 115)(105 111 116)(106 112 117)(107 113 118)(108 114 119)(121 127 138)(122 128 133)(123 129 134)(124 130 135)(125 131 136)(126 132 137)(139 145 156)(140 146 151)(141 147 152)(142 148 153)(143 149 154)(144 150 155)(157 163 174)(158 164 169)(159 165 170)(160 166 171)(161 167 172)(162 168 173)
(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)
(1 120 4 117)(2 119 5 116)(3 118 6 115)(7 101 10 98)(8 100 11 97)(9 99 12 102)(13 96 16 93)(14 95 17 92)(15 94 18 91)(19 113 22 110)(20 112 23 109)(21 111 24 114)(25 107 28 104)(26 106 29 103)(27 105 30 108)(31 123 34 126)(32 122 35 125)(33 121 36 124)(37 131 40 128)(38 130 41 127)(39 129 42 132)(43 133 46 136)(44 138 47 135)(45 137 48 134)(49 141 52 144)(50 140 53 143)(51 139 54 142)(55 149 58 146)(56 148 59 145)(57 147 60 150)(61 151 64 154)(62 156 65 153)(63 155 66 152)(67 159 70 162)(68 158 71 161)(69 157 72 160)(73 167 76 164)(74 166 77 163)(75 165 78 168)(79 169 82 172)(80 174 83 171)(81 173 84 170)(85 177 88 180)(86 176 89 179)(87 175 90 178)
G:=sub<Sym(180)| (1,98,80,62,44)(2,99,81,63,45)(3,100,82,64,46)(4,101,83,65,47)(5,102,84,66,48)(6,97,79,61,43)(7,174,156,138,120)(8,169,151,133,115)(9,170,152,134,116)(10,171,153,135,117)(11,172,154,136,118)(12,173,155,137,119)(13,168,150,132,114)(14,163,145,127,109)(15,164,146,128,110)(16,165,147,129,111)(17,166,148,130,112)(18,167,149,131,113)(19,94,73,55,37)(20,95,74,56,38)(21,96,75,57,39)(22,91,76,58,40)(23,92,77,59,41)(24,93,78,60,42)(25,89,71,53,35)(26,90,72,54,36)(27,85,67,49,31)(28,86,68,50,32)(29,87,69,51,33)(30,88,70,52,34)(103,175,157,139,121)(104,176,158,140,122)(105,177,159,141,123)(106,178,160,142,124)(107,179,161,143,125)(108,180,162,144,126), (1,23,29)(2,24,30)(3,19,25)(4,20,26)(5,21,27)(6,22,28)(7,175,14)(8,176,15)(9,177,16)(10,178,17)(11,179,18)(12,180,13)(31,48,39)(32,43,40)(33,44,41)(34,45,42)(35,46,37)(36,47,38)(49,66,57)(50,61,58)(51,62,59)(52,63,60)(53,64,55)(54,65,56)(67,84,75)(68,79,76)(69,80,77)(70,81,78)(71,82,73)(72,83,74)(85,102,96)(86,97,91)(87,98,92)(88,99,93)(89,100,94)(90,101,95)(103,109,120)(104,110,115)(105,111,116)(106,112,117)(107,113,118)(108,114,119)(121,127,138)(122,128,133)(123,129,134)(124,130,135)(125,131,136)(126,132,137)(139,145,156)(140,146,151)(141,147,152)(142,148,153)(143,149,154)(144,150,155)(157,163,174)(158,164,169)(159,165,170)(160,166,171)(161,167,172)(162,168,173), (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), (1,120,4,117)(2,119,5,116)(3,118,6,115)(7,101,10,98)(8,100,11,97)(9,99,12,102)(13,96,16,93)(14,95,17,92)(15,94,18,91)(19,113,22,110)(20,112,23,109)(21,111,24,114)(25,107,28,104)(26,106,29,103)(27,105,30,108)(31,123,34,126)(32,122,35,125)(33,121,36,124)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,133,46,136)(44,138,47,135)(45,137,48,134)(49,141,52,144)(50,140,53,143)(51,139,54,142)(55,149,58,146)(56,148,59,145)(57,147,60,150)(61,151,64,154)(62,156,65,153)(63,155,66,152)(67,159,70,162)(68,158,71,161)(69,157,72,160)(73,167,76,164)(74,166,77,163)(75,165,78,168)(79,169,82,172)(80,174,83,171)(81,173,84,170)(85,177,88,180)(86,176,89,179)(87,175,90,178)>;
G:=Group( (1,98,80,62,44)(2,99,81,63,45)(3,100,82,64,46)(4,101,83,65,47)(5,102,84,66,48)(6,97,79,61,43)(7,174,156,138,120)(8,169,151,133,115)(9,170,152,134,116)(10,171,153,135,117)(11,172,154,136,118)(12,173,155,137,119)(13,168,150,132,114)(14,163,145,127,109)(15,164,146,128,110)(16,165,147,129,111)(17,166,148,130,112)(18,167,149,131,113)(19,94,73,55,37)(20,95,74,56,38)(21,96,75,57,39)(22,91,76,58,40)(23,92,77,59,41)(24,93,78,60,42)(25,89,71,53,35)(26,90,72,54,36)(27,85,67,49,31)(28,86,68,50,32)(29,87,69,51,33)(30,88,70,52,34)(103,175,157,139,121)(104,176,158,140,122)(105,177,159,141,123)(106,178,160,142,124)(107,179,161,143,125)(108,180,162,144,126), (1,23,29)(2,24,30)(3,19,25)(4,20,26)(5,21,27)(6,22,28)(7,175,14)(8,176,15)(9,177,16)(10,178,17)(11,179,18)(12,180,13)(31,48,39)(32,43,40)(33,44,41)(34,45,42)(35,46,37)(36,47,38)(49,66,57)(50,61,58)(51,62,59)(52,63,60)(53,64,55)(54,65,56)(67,84,75)(68,79,76)(69,80,77)(70,81,78)(71,82,73)(72,83,74)(85,102,96)(86,97,91)(87,98,92)(88,99,93)(89,100,94)(90,101,95)(103,109,120)(104,110,115)(105,111,116)(106,112,117)(107,113,118)(108,114,119)(121,127,138)(122,128,133)(123,129,134)(124,130,135)(125,131,136)(126,132,137)(139,145,156)(140,146,151)(141,147,152)(142,148,153)(143,149,154)(144,150,155)(157,163,174)(158,164,169)(159,165,170)(160,166,171)(161,167,172)(162,168,173), (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), (1,120,4,117)(2,119,5,116)(3,118,6,115)(7,101,10,98)(8,100,11,97)(9,99,12,102)(13,96,16,93)(14,95,17,92)(15,94,18,91)(19,113,22,110)(20,112,23,109)(21,111,24,114)(25,107,28,104)(26,106,29,103)(27,105,30,108)(31,123,34,126)(32,122,35,125)(33,121,36,124)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,133,46,136)(44,138,47,135)(45,137,48,134)(49,141,52,144)(50,140,53,143)(51,139,54,142)(55,149,58,146)(56,148,59,145)(57,147,60,150)(61,151,64,154)(62,156,65,153)(63,155,66,152)(67,159,70,162)(68,158,71,161)(69,157,72,160)(73,167,76,164)(74,166,77,163)(75,165,78,168)(79,169,82,172)(80,174,83,171)(81,173,84,170)(85,177,88,180)(86,176,89,179)(87,175,90,178) );
G=PermutationGroup([[(1,98,80,62,44),(2,99,81,63,45),(3,100,82,64,46),(4,101,83,65,47),(5,102,84,66,48),(6,97,79,61,43),(7,174,156,138,120),(8,169,151,133,115),(9,170,152,134,116),(10,171,153,135,117),(11,172,154,136,118),(12,173,155,137,119),(13,168,150,132,114),(14,163,145,127,109),(15,164,146,128,110),(16,165,147,129,111),(17,166,148,130,112),(18,167,149,131,113),(19,94,73,55,37),(20,95,74,56,38),(21,96,75,57,39),(22,91,76,58,40),(23,92,77,59,41),(24,93,78,60,42),(25,89,71,53,35),(26,90,72,54,36),(27,85,67,49,31),(28,86,68,50,32),(29,87,69,51,33),(30,88,70,52,34),(103,175,157,139,121),(104,176,158,140,122),(105,177,159,141,123),(106,178,160,142,124),(107,179,161,143,125),(108,180,162,144,126)], [(1,23,29),(2,24,30),(3,19,25),(4,20,26),(5,21,27),(6,22,28),(7,175,14),(8,176,15),(9,177,16),(10,178,17),(11,179,18),(12,180,13),(31,48,39),(32,43,40),(33,44,41),(34,45,42),(35,46,37),(36,47,38),(49,66,57),(50,61,58),(51,62,59),(52,63,60),(53,64,55),(54,65,56),(67,84,75),(68,79,76),(69,80,77),(70,81,78),(71,82,73),(72,83,74),(85,102,96),(86,97,91),(87,98,92),(88,99,93),(89,100,94),(90,101,95),(103,109,120),(104,110,115),(105,111,116),(106,112,117),(107,113,118),(108,114,119),(121,127,138),(122,128,133),(123,129,134),(124,130,135),(125,131,136),(126,132,137),(139,145,156),(140,146,151),(141,147,152),(142,148,153),(143,149,154),(144,150,155),(157,163,174),(158,164,169),(159,165,170),(160,166,171),(161,167,172),(162,168,173)], [(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)], [(1,120,4,117),(2,119,5,116),(3,118,6,115),(7,101,10,98),(8,100,11,97),(9,99,12,102),(13,96,16,93),(14,95,17,92),(15,94,18,91),(19,113,22,110),(20,112,23,109),(21,111,24,114),(25,107,28,104),(26,106,29,103),(27,105,30,108),(31,123,34,126),(32,122,35,125),(33,121,36,124),(37,131,40,128),(38,130,41,127),(39,129,42,132),(43,133,46,136),(44,138,47,135),(45,137,48,134),(49,141,52,144),(50,140,53,143),(51,139,54,142),(55,149,58,146),(56,148,59,145),(57,147,60,150),(61,151,64,154),(62,156,65,153),(63,155,66,152),(67,159,70,162),(68,158,71,161),(69,157,72,160),(73,167,76,164),(74,166,77,163),(75,165,78,168),(79,169,82,172),(80,174,83,171),(81,173,84,170),(85,177,88,180),(86,176,89,179),(87,175,90,178)]])
C5×C3⋊Dic3 is a maximal subgroup of
(C3×C15)⋊9C8 C30.D6 C32⋊7D20 C15⋊Dic6 C5×S3×Dic3 D30.S3 C32⋊3D20 C32⋊3Dic10 C3⋊S3×C20
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 15A | ··· | 15P | 20A | ··· | 20H | 30A | ··· | 30P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | S3 | Dic3 | C5×S3 | C5×Dic3 |
| kernel | C5×C3⋊Dic3 | C3×C30 | C3×C15 | C3⋊Dic3 | C3×C6 | C32 | C30 | C15 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 4 | 16 | 16 |
Matrix representation of C5×C3⋊Dic3 ►in GL4(𝔽61) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 1 |
| 0 | 0 | 60 | 0 |
| 1 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 |
| 0 | 0 | 60 | 0 |
| 49 | 20 | 0 | 0 |
| 8 | 12 | 0 | 0 |
| 0 | 0 | 48 | 22 |
| 0 | 0 | 9 | 13 |
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,60,60,0,0,1,0],[1,1,0,0,60,0,0,0,0,0,60,60,0,0,1,0],[49,8,0,0,20,12,0,0,0,0,48,9,0,0,22,13] >;
C5×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_5\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C5xC3:Dic3"); // GroupNames label
G:=SmallGroup(180,16);
// by ID
G=gap.SmallGroup(180,16);
# by ID
G:=PCGroup([5,-2,-5,-2,-3,-3,50,803,3004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export