direct product, metabelian, supersoluble, monomial
Aliases: C3×C6×Dic6, C12.11C62, C62.167D6, Dic3.1C62, C6⋊(Q8×C32), C32⋊8(C6×Q8), (C32×C6)⋊6Q8, C33⋊16(C2×Q8), (C6×C12).38C6, (C6×C12).58S3, C6.1(C2×C62), C2.3(S3×C62), C12.115(S3×C6), (C3×C12).234D6, (C2×C6).19C62, C62.70(C2×C6), (C6×Dic3).14C6, (C32×C6).75C23, (C3×C62).56C22, (C32×C12).85C22, (C32×Dic3).31C22, C3⋊1(Q8×C3×C6), C4.11(S3×C3×C6), C6.73(S3×C2×C6), (C3×C6)⋊5(C3×Q8), (C3×C6×C12).11C2, C22.8(S3×C3×C6), (C2×C6).95(S3×C6), (C2×C12).9(C3×C6), (C2×C12).47(C3×S3), (C3×C12).86(C2×C6), (C2×C4).4(S3×C32), (Dic3×C3×C6).12C2, (C3×C6).49(C22×C6), (C2×Dic3).3(C3×C6), (C3×C6).194(C22×S3), (C3×Dic3).16(C2×C6), SmallGroup(432,700)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C6×Dic6
G = < a,b,c,d | a3=b6=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 488 in 292 conjugacy classes, 162 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C62, C62, C62, C2×Dic6, C6×Q8, C32×C6, C32×C6, C3×Dic6, C6×Dic3, C6×C12, C6×C12, C6×C12, Q8×C32, C32×Dic3, C32×C12, C3×C62, C6×Dic6, Q8×C3×C6, C32×Dic6, Dic3×C3×C6, C3×C6×C12, C3×C6×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, C32, D6, C2×C6, C2×Q8, C3×S3, C3×C6, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C62, C2×Dic6, C6×Q8, S3×C32, C3×Dic6, Q8×C32, S3×C2×C6, C2×C62, S3×C3×C6, C6×Dic6, Q8×C3×C6, C32×Dic6, S3×C62, C3×C6×Dic6
►On 144 points
Generators in S144
(1 131 16)(2 132 17)(3 121 18)(4 122 19)(5 123 20)(6 124 21)(7 125 22)(8 126 23)(9 127 24)(10 128 13)(11 129 14)(12 130 15)(25 42 96)(26 43 85)(27 44 86)(28 45 87)(29 46 88)(30 47 89)(31 48 90)(32 37 91)(33 38 92)(34 39 93)(35 40 94)(36 41 95)(49 72 107)(50 61 108)(51 62 97)(52 63 98)(53 64 99)(54 65 100)(55 66 101)(56 67 102)(57 68 103)(58 69 104)(59 70 105)(60 71 106)(73 136 116)(74 137 117)(75 138 118)(76 139 119)(77 140 120)(78 141 109)(79 142 110)(80 143 111)(81 144 112)(82 133 113)(83 134 114)(84 135 115)
(1 99 9 107 5 103)(2 100 10 108 6 104)(3 101 11 97 7 105)(4 102 12 98 8 106)(13 61 21 69 17 65)(14 62 22 70 18 66)(15 63 23 71 19 67)(16 64 24 72 20 68)(25 117 29 109 33 113)(26 118 30 110 34 114)(27 119 31 111 35 115)(28 120 32 112 36 116)(37 81 41 73 45 77)(38 82 42 74 46 78)(39 83 43 75 47 79)(40 84 44 76 48 80)(49 123 57 131 53 127)(50 124 58 132 54 128)(51 125 59 121 55 129)(52 126 60 122 56 130)(85 138 89 142 93 134)(86 139 90 143 94 135)(87 140 91 144 95 136)(88 141 92 133 96 137)
(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)
(1 93 7 87)(2 92 8 86)(3 91 9 85)(4 90 10 96)(5 89 11 95)(6 88 12 94)(13 42 19 48)(14 41 20 47)(15 40 21 46)(16 39 22 45)(17 38 23 44)(18 37 24 43)(25 122 31 128)(26 121 32 127)(27 132 33 126)(28 131 34 125)(29 130 35 124)(30 129 36 123)(49 118 55 112)(50 117 56 111)(51 116 57 110)(52 115 58 109)(53 114 59 120)(54 113 60 119)(61 74 67 80)(62 73 68 79)(63 84 69 78)(64 83 70 77)(65 82 71 76)(66 81 72 75)(97 136 103 142)(98 135 104 141)(99 134 105 140)(100 133 106 139)(101 144 107 138)(102 143 108 137)
G:=sub<Sym(144)| (1,131,16)(2,132,17)(3,121,18)(4,122,19)(5,123,20)(6,124,21)(7,125,22)(8,126,23)(9,127,24)(10,128,13)(11,129,14)(12,130,15)(25,42,96)(26,43,85)(27,44,86)(28,45,87)(29,46,88)(30,47,89)(31,48,90)(32,37,91)(33,38,92)(34,39,93)(35,40,94)(36,41,95)(49,72,107)(50,61,108)(51,62,97)(52,63,98)(53,64,99)(54,65,100)(55,66,101)(56,67,102)(57,68,103)(58,69,104)(59,70,105)(60,71,106)(73,136,116)(74,137,117)(75,138,118)(76,139,119)(77,140,120)(78,141,109)(79,142,110)(80,143,111)(81,144,112)(82,133,113)(83,134,114)(84,135,115), (1,99,9,107,5,103)(2,100,10,108,6,104)(3,101,11,97,7,105)(4,102,12,98,8,106)(13,61,21,69,17,65)(14,62,22,70,18,66)(15,63,23,71,19,67)(16,64,24,72,20,68)(25,117,29,109,33,113)(26,118,30,110,34,114)(27,119,31,111,35,115)(28,120,32,112,36,116)(37,81,41,73,45,77)(38,82,42,74,46,78)(39,83,43,75,47,79)(40,84,44,76,48,80)(49,123,57,131,53,127)(50,124,58,132,54,128)(51,125,59,121,55,129)(52,126,60,122,56,130)(85,138,89,142,93,134)(86,139,90,143,94,135)(87,140,91,144,95,136)(88,141,92,133,96,137), (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), (1,93,7,87)(2,92,8,86)(3,91,9,85)(4,90,10,96)(5,89,11,95)(6,88,12,94)(13,42,19,48)(14,41,20,47)(15,40,21,46)(16,39,22,45)(17,38,23,44)(18,37,24,43)(25,122,31,128)(26,121,32,127)(27,132,33,126)(28,131,34,125)(29,130,35,124)(30,129,36,123)(49,118,55,112)(50,117,56,111)(51,116,57,110)(52,115,58,109)(53,114,59,120)(54,113,60,119)(61,74,67,80)(62,73,68,79)(63,84,69,78)(64,83,70,77)(65,82,71,76)(66,81,72,75)(97,136,103,142)(98,135,104,141)(99,134,105,140)(100,133,106,139)(101,144,107,138)(102,143,108,137)>;
G:=Group( (1,131,16)(2,132,17)(3,121,18)(4,122,19)(5,123,20)(6,124,21)(7,125,22)(8,126,23)(9,127,24)(10,128,13)(11,129,14)(12,130,15)(25,42,96)(26,43,85)(27,44,86)(28,45,87)(29,46,88)(30,47,89)(31,48,90)(32,37,91)(33,38,92)(34,39,93)(35,40,94)(36,41,95)(49,72,107)(50,61,108)(51,62,97)(52,63,98)(53,64,99)(54,65,100)(55,66,101)(56,67,102)(57,68,103)(58,69,104)(59,70,105)(60,71,106)(73,136,116)(74,137,117)(75,138,118)(76,139,119)(77,140,120)(78,141,109)(79,142,110)(80,143,111)(81,144,112)(82,133,113)(83,134,114)(84,135,115), (1,99,9,107,5,103)(2,100,10,108,6,104)(3,101,11,97,7,105)(4,102,12,98,8,106)(13,61,21,69,17,65)(14,62,22,70,18,66)(15,63,23,71,19,67)(16,64,24,72,20,68)(25,117,29,109,33,113)(26,118,30,110,34,114)(27,119,31,111,35,115)(28,120,32,112,36,116)(37,81,41,73,45,77)(38,82,42,74,46,78)(39,83,43,75,47,79)(40,84,44,76,48,80)(49,123,57,131,53,127)(50,124,58,132,54,128)(51,125,59,121,55,129)(52,126,60,122,56,130)(85,138,89,142,93,134)(86,139,90,143,94,135)(87,140,91,144,95,136)(88,141,92,133,96,137), (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), (1,93,7,87)(2,92,8,86)(3,91,9,85)(4,90,10,96)(5,89,11,95)(6,88,12,94)(13,42,19,48)(14,41,20,47)(15,40,21,46)(16,39,22,45)(17,38,23,44)(18,37,24,43)(25,122,31,128)(26,121,32,127)(27,132,33,126)(28,131,34,125)(29,130,35,124)(30,129,36,123)(49,118,55,112)(50,117,56,111)(51,116,57,110)(52,115,58,109)(53,114,59,120)(54,113,60,119)(61,74,67,80)(62,73,68,79)(63,84,69,78)(64,83,70,77)(65,82,71,76)(66,81,72,75)(97,136,103,142)(98,135,104,141)(99,134,105,140)(100,133,106,139)(101,144,107,138)(102,143,108,137) );
G=PermutationGroup([[(1,131,16),(2,132,17),(3,121,18),(4,122,19),(5,123,20),(6,124,21),(7,125,22),(8,126,23),(9,127,24),(10,128,13),(11,129,14),(12,130,15),(25,42,96),(26,43,85),(27,44,86),(28,45,87),(29,46,88),(30,47,89),(31,48,90),(32,37,91),(33,38,92),(34,39,93),(35,40,94),(36,41,95),(49,72,107),(50,61,108),(51,62,97),(52,63,98),(53,64,99),(54,65,100),(55,66,101),(56,67,102),(57,68,103),(58,69,104),(59,70,105),(60,71,106),(73,136,116),(74,137,117),(75,138,118),(76,139,119),(77,140,120),(78,141,109),(79,142,110),(80,143,111),(81,144,112),(82,133,113),(83,134,114),(84,135,115)], [(1,99,9,107,5,103),(2,100,10,108,6,104),(3,101,11,97,7,105),(4,102,12,98,8,106),(13,61,21,69,17,65),(14,62,22,70,18,66),(15,63,23,71,19,67),(16,64,24,72,20,68),(25,117,29,109,33,113),(26,118,30,110,34,114),(27,119,31,111,35,115),(28,120,32,112,36,116),(37,81,41,73,45,77),(38,82,42,74,46,78),(39,83,43,75,47,79),(40,84,44,76,48,80),(49,123,57,131,53,127),(50,124,58,132,54,128),(51,125,59,121,55,129),(52,126,60,122,56,130),(85,138,89,142,93,134),(86,139,90,143,94,135),(87,140,91,144,95,136),(88,141,92,133,96,137)], [(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)], [(1,93,7,87),(2,92,8,86),(3,91,9,85),(4,90,10,96),(5,89,11,95),(6,88,12,94),(13,42,19,48),(14,41,20,47),(15,40,21,46),(16,39,22,45),(17,38,23,44),(18,37,24,43),(25,122,31,128),(26,121,32,127),(27,132,33,126),(28,131,34,125),(29,130,35,124),(30,129,36,123),(49,118,55,112),(50,117,56,111),(51,116,57,110),(52,115,58,109),(53,114,59,120),(54,113,60,119),(61,74,67,80),(62,73,68,79),(63,84,69,78),(64,83,70,77),(65,82,71,76),(66,81,72,75),(97,136,103,142),(98,135,104,141),(99,134,105,140),(100,133,106,139),(101,144,107,138),(102,143,108,137)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6X | 6Y | ··· | 6AY | 12A | ··· | 12AZ | 12BA | ··· | 12CF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | S3×C6 | C3×Dic6 |
| kernel | C3×C6×Dic6 | C32×Dic6 | Dic3×C3×C6 | C3×C6×C12 | C6×Dic6 | C3×Dic6 | C6×Dic3 | C6×C12 | C6×C12 | C32×C6 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 2 | 1 | 8 | 32 | 16 | 8 | 1 | 2 | 2 | 1 | 8 | 4 | 16 | 16 | 8 | 32 |
Matrix representation of C3×C6×Dic6 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 12 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [3,0,0,0,1,0,0,0,1],[4,0,0,0,4,0,0,0,4],[12,0,0,0,7,0,0,0,2],[1,0,0,0,0,12,0,1,0] >;
C3×C6×Dic6 in GAP, Magma, Sage, TeX
C_3\times C_6\times {\rm Dic}_6 % in TeX
G:=Group("C3xC6xDic6"); // GroupNames label
G:=SmallGroup(432,700);
// by ID
G=gap.SmallGroup(432,700);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,504,1598,394,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations