direct product, metabelian, supersoluble, monomial
Aliases: C3×C6.11D12, C62.143D6, (C6×C12)⋊10S3, (C6×C12)⋊10C6, C6.27(S3×C12), C6.23(C3×D12), (C3×C6).65D12, C62.67(C2×C6), (C32×C6).72D4, C32⋊12(D6⋊C4), C33⋊15(C22⋊C4), C6.26(C12⋊S3), C6.34(C32⋊7D4), (C3×C62).49C22, (C3×C6×C12)⋊2C2, (C6×C3⋊S3)⋊4C4, C3⋊2(C3×D6⋊C4), (C2×C3⋊S3)⋊4C12, (C2×C12)⋊2(C3×S3), C6.28(C4×C3⋊S3), C2.5(C12×C3⋊S3), (C2×C12)⋊1(C3⋊S3), (C2×C6).70(S3×C6), (C3×C6).79(C4×S3), (C6×C3⋊Dic3)⋊5C2, (C2×C3⋊Dic3)⋊8C6, (C3×C6).68(C3×D4), C6.37(C3×C3⋊D4), C22.6(C6×C3⋊S3), C2.2(C3×C12⋊S3), (C3×C6).50(C2×C12), (C22×C3⋊S3).5C6, C32⋊9(C3×C22⋊C4), C2.2(C3×C32⋊7D4), (C32×C6).58(C2×C4), (C3×C6).107(C3⋊D4), (C2×C4)⋊1(C3×C3⋊S3), (C2×C6×C3⋊S3).4C2, (C2×C6).64(C2×C3⋊S3), SmallGroup(432,490)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C6.11D12
G = < a,b,c,d | a3=b6=c12=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 884 in 268 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, D6⋊C4, C3×C22⋊C4, C3×C3⋊S3, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C3×D6⋊C4, C6.11D12, C6×C3⋊Dic3, C3×C6×C12, C2×C6×C3⋊S3, C3×C6.11D12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, S3×C6, C2×C3⋊S3, D6⋊C4, C3×C22⋊C4, C3×C3⋊S3, S3×C12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C3⋊S3, C3×D6⋊C4, C6.11D12, C12×C3⋊S3, C3×C12⋊S3, C3×C32⋊7D4, C3×C6.11D12
►On 144 points
Generators in S144
(1 107 134)(2 108 135)(3 97 136)(4 98 137)(5 99 138)(6 100 139)(7 101 140)(8 102 141)(9 103 142)(10 104 143)(11 105 144)(12 106 133)(13 58 111)(14 59 112)(15 60 113)(16 49 114)(17 50 115)(18 51 116)(19 52 117)(20 53 118)(21 54 119)(22 55 120)(23 56 109)(24 57 110)(25 121 78)(26 122 79)(27 123 80)(28 124 81)(29 125 82)(30 126 83)(31 127 84)(32 128 73)(33 129 74)(34 130 75)(35 131 76)(36 132 77)(37 87 68)(38 88 69)(39 89 70)(40 90 71)(41 91 72)(42 92 61)(43 93 62)(44 94 63)(45 95 64)(46 96 65)(47 85 66)(48 86 67)
(1 55 142 18 99 112)(2 56 143 19 100 113)(3 57 144 20 101 114)(4 58 133 21 102 115)(5 59 134 22 103 116)(6 60 135 23 104 117)(7 49 136 24 105 118)(8 50 137 13 106 119)(9 51 138 14 107 120)(10 52 139 15 108 109)(11 53 140 16 97 110)(12 54 141 17 98 111)(25 88 129 65 82 42)(26 89 130 66 83 43)(27 90 131 67 84 44)(28 91 132 68 73 45)(29 92 121 69 74 46)(30 93 122 70 75 47)(31 94 123 71 76 48)(32 95 124 72 77 37)(33 96 125 61 78 38)(34 85 126 62 79 39)(35 86 127 63 80 40)(36 87 128 64 81 41)
(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 63 18 35)(2 34 19 62)(3 61 20 33)(4 32 21 72)(5 71 22 31)(6 30 23 70)(7 69 24 29)(8 28 13 68)(9 67 14 27)(10 26 15 66)(11 65 16 25)(12 36 17 64)(37 102 124 58)(38 57 125 101)(39 100 126 56)(40 55 127 99)(41 98 128 54)(42 53 129 97)(43 108 130 52)(44 51 131 107)(45 106 132 50)(46 49 121 105)(47 104 122 60)(48 59 123 103)(73 119 91 137)(74 136 92 118)(75 117 93 135)(76 134 94 116)(77 115 95 133)(78 144 96 114)(79 113 85 143)(80 142 86 112)(81 111 87 141)(82 140 88 110)(83 109 89 139)(84 138 90 120)
G:=sub<Sym(144)| (1,107,134)(2,108,135)(3,97,136)(4,98,137)(5,99,138)(6,100,139)(7,101,140)(8,102,141)(9,103,142)(10,104,143)(11,105,144)(12,106,133)(13,58,111)(14,59,112)(15,60,113)(16,49,114)(17,50,115)(18,51,116)(19,52,117)(20,53,118)(21,54,119)(22,55,120)(23,56,109)(24,57,110)(25,121,78)(26,122,79)(27,123,80)(28,124,81)(29,125,82)(30,126,83)(31,127,84)(32,128,73)(33,129,74)(34,130,75)(35,131,76)(36,132,77)(37,87,68)(38,88,69)(39,89,70)(40,90,71)(41,91,72)(42,92,61)(43,93,62)(44,94,63)(45,95,64)(46,96,65)(47,85,66)(48,86,67), (1,55,142,18,99,112)(2,56,143,19,100,113)(3,57,144,20,101,114)(4,58,133,21,102,115)(5,59,134,22,103,116)(6,60,135,23,104,117)(7,49,136,24,105,118)(8,50,137,13,106,119)(9,51,138,14,107,120)(10,52,139,15,108,109)(11,53,140,16,97,110)(12,54,141,17,98,111)(25,88,129,65,82,42)(26,89,130,66,83,43)(27,90,131,67,84,44)(28,91,132,68,73,45)(29,92,121,69,74,46)(30,93,122,70,75,47)(31,94,123,71,76,48)(32,95,124,72,77,37)(33,96,125,61,78,38)(34,85,126,62,79,39)(35,86,127,63,80,40)(36,87,128,64,81,41), (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), (1,63,18,35)(2,34,19,62)(3,61,20,33)(4,32,21,72)(5,71,22,31)(6,30,23,70)(7,69,24,29)(8,28,13,68)(9,67,14,27)(10,26,15,66)(11,65,16,25)(12,36,17,64)(37,102,124,58)(38,57,125,101)(39,100,126,56)(40,55,127,99)(41,98,128,54)(42,53,129,97)(43,108,130,52)(44,51,131,107)(45,106,132,50)(46,49,121,105)(47,104,122,60)(48,59,123,103)(73,119,91,137)(74,136,92,118)(75,117,93,135)(76,134,94,116)(77,115,95,133)(78,144,96,114)(79,113,85,143)(80,142,86,112)(81,111,87,141)(82,140,88,110)(83,109,89,139)(84,138,90,120)>;
G:=Group( (1,107,134)(2,108,135)(3,97,136)(4,98,137)(5,99,138)(6,100,139)(7,101,140)(8,102,141)(9,103,142)(10,104,143)(11,105,144)(12,106,133)(13,58,111)(14,59,112)(15,60,113)(16,49,114)(17,50,115)(18,51,116)(19,52,117)(20,53,118)(21,54,119)(22,55,120)(23,56,109)(24,57,110)(25,121,78)(26,122,79)(27,123,80)(28,124,81)(29,125,82)(30,126,83)(31,127,84)(32,128,73)(33,129,74)(34,130,75)(35,131,76)(36,132,77)(37,87,68)(38,88,69)(39,89,70)(40,90,71)(41,91,72)(42,92,61)(43,93,62)(44,94,63)(45,95,64)(46,96,65)(47,85,66)(48,86,67), (1,55,142,18,99,112)(2,56,143,19,100,113)(3,57,144,20,101,114)(4,58,133,21,102,115)(5,59,134,22,103,116)(6,60,135,23,104,117)(7,49,136,24,105,118)(8,50,137,13,106,119)(9,51,138,14,107,120)(10,52,139,15,108,109)(11,53,140,16,97,110)(12,54,141,17,98,111)(25,88,129,65,82,42)(26,89,130,66,83,43)(27,90,131,67,84,44)(28,91,132,68,73,45)(29,92,121,69,74,46)(30,93,122,70,75,47)(31,94,123,71,76,48)(32,95,124,72,77,37)(33,96,125,61,78,38)(34,85,126,62,79,39)(35,86,127,63,80,40)(36,87,128,64,81,41), (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), (1,63,18,35)(2,34,19,62)(3,61,20,33)(4,32,21,72)(5,71,22,31)(6,30,23,70)(7,69,24,29)(8,28,13,68)(9,67,14,27)(10,26,15,66)(11,65,16,25)(12,36,17,64)(37,102,124,58)(38,57,125,101)(39,100,126,56)(40,55,127,99)(41,98,128,54)(42,53,129,97)(43,108,130,52)(44,51,131,107)(45,106,132,50)(46,49,121,105)(47,104,122,60)(48,59,123,103)(73,119,91,137)(74,136,92,118)(75,117,93,135)(76,134,94,116)(77,115,95,133)(78,144,96,114)(79,113,85,143)(80,142,86,112)(81,111,87,141)(82,140,88,110)(83,109,89,139)(84,138,90,120) );
G=PermutationGroup([[(1,107,134),(2,108,135),(3,97,136),(4,98,137),(5,99,138),(6,100,139),(7,101,140),(8,102,141),(9,103,142),(10,104,143),(11,105,144),(12,106,133),(13,58,111),(14,59,112),(15,60,113),(16,49,114),(17,50,115),(18,51,116),(19,52,117),(20,53,118),(21,54,119),(22,55,120),(23,56,109),(24,57,110),(25,121,78),(26,122,79),(27,123,80),(28,124,81),(29,125,82),(30,126,83),(31,127,84),(32,128,73),(33,129,74),(34,130,75),(35,131,76),(36,132,77),(37,87,68),(38,88,69),(39,89,70),(40,90,71),(41,91,72),(42,92,61),(43,93,62),(44,94,63),(45,95,64),(46,96,65),(47,85,66),(48,86,67)], [(1,55,142,18,99,112),(2,56,143,19,100,113),(3,57,144,20,101,114),(4,58,133,21,102,115),(5,59,134,22,103,116),(6,60,135,23,104,117),(7,49,136,24,105,118),(8,50,137,13,106,119),(9,51,138,14,107,120),(10,52,139,15,108,109),(11,53,140,16,97,110),(12,54,141,17,98,111),(25,88,129,65,82,42),(26,89,130,66,83,43),(27,90,131,67,84,44),(28,91,132,68,73,45),(29,92,121,69,74,46),(30,93,122,70,75,47),(31,94,123,71,76,48),(32,95,124,72,77,37),(33,96,125,61,78,38),(34,85,126,62,79,39),(35,86,127,63,80,40),(36,87,128,64,81,41)], [(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)], [(1,63,18,35),(2,34,19,62),(3,61,20,33),(4,32,21,72),(5,71,22,31),(6,30,23,70),(7,69,24,29),(8,28,13,68),(9,67,14,27),(10,26,15,66),(11,65,16,25),(12,36,17,64),(37,102,124,58),(38,57,125,101),(39,100,126,56),(40,55,127,99),(41,98,128,54),(42,53,129,97),(43,108,130,52),(44,51,131,107),(45,106,132,50),(46,49,121,105),(47,104,122,60),(48,59,123,103),(73,119,91,137),(74,136,92,118),(75,117,93,135),(76,134,94,116),(77,115,95,133),(78,144,96,114),(79,113,85,143),(80,142,86,112),(81,111,87,141),(82,140,88,110),(83,109,89,139),(84,138,90,120)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6AP | 6AQ | 6AR | 6AS | 6AT | 12A | ··· | 12AZ | 12BA | 12BB | 12BC | 12BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 18 | 18 | 18 | 18 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | C3×S3 | C4×S3 | D12 | C3⋊D4 | C3×D4 | S3×C6 | S3×C12 | C3×D12 | C3×C3⋊D4 |
| kernel | C3×C6.11D12 | C6×C3⋊Dic3 | C3×C6×C12 | C2×C6×C3⋊S3 | C6.11D12 | C6×C3⋊S3 | C2×C3⋊Dic3 | C6×C12 | C22×C3⋊S3 | C2×C3⋊S3 | C6×C12 | C32×C6 | C62 | C2×C12 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 4 | 2 | 4 | 8 | 8 | 8 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C3×C6.11D12 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,9,0] >;
C3×C6.11D12 in GAP, Magma, Sage, TeX
C_3\times C_6._{11}D_{12} % in TeX
G:=Group("C3xC6.11D12"); // GroupNames label
G:=SmallGroup(432,490);
// by ID
G=gap.SmallGroup(432,490);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=b^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=b^3*c^-1>;
// generators/relations