direct product, metabelian, supersoluble, monomial
Aliases: C3⋊S3×Dic6, C12.38S32, C3⋊3(S3×Dic6), C32⋊8(S3×Q8), C33⋊10(C2×Q8), (C3×Dic6)⋊6S3, C33⋊8Q8⋊7C2, C33⋊4Q8⋊6C2, (C3×C12).141D6, C3⋊Dic3.44D6, (C3×Dic3).13D6, C32⋊13(C2×Dic6), (C32×Dic6)⋊10C2, (C32×C6).41C23, C33⋊5C4.9C22, (C32×C12).43C22, (C32×Dic3).13C22, C3⋊2(Q8×C3⋊S3), C6.51(C2×S32), (C3×C3⋊S3)⋊5Q8, (C4×C3⋊S3).4S3, C4.13(S3×C3⋊S3), (C12×C3⋊S3).6C2, C12.35(C2×C3⋊S3), (C2×C3⋊S3).51D6, C6.4(C22×C3⋊S3), Dic3.1(C2×C3⋊S3), (Dic3×C3⋊S3).2C2, (C6×C3⋊S3).51C22, (C3×C6).143(C22×S3), (C3×C3⋊Dic3).42C22, C2.8(C2×S3×C3⋊S3), SmallGroup(432,663)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊S3×Dic6
G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=d6, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 1368 in 276 conjugacy classes, 72 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C2×Dic6, S3×Q8, C3×C3⋊S3, C32×C6, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, C32×C12, C6×C3⋊S3, S3×Dic6, Q8×C3⋊S3, Dic3×C3⋊S3, C33⋊4Q8, C32×Dic6, C12×C3⋊S3, C33⋊8Q8, C3⋊S3×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, Dic6, C22×S3, S32, C2×C3⋊S3, C2×Dic6, S3×Q8, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×Dic6, Q8×C3⋊S3, C2×S3×C3⋊S3, C3⋊S3×Dic6
►On 144 points
Generators in S144
(1 20 138)(2 21 139)(3 22 140)(4 23 141)(5 24 142)(6 13 143)(7 14 144)(8 15 133)(9 16 134)(10 17 135)(11 18 136)(12 19 137)(25 109 82)(26 110 83)(27 111 84)(28 112 73)(29 113 74)(30 114 75)(31 115 76)(32 116 77)(33 117 78)(34 118 79)(35 119 80)(36 120 81)(37 49 96)(38 50 85)(39 51 86)(40 52 87)(41 53 88)(42 54 89)(43 55 90)(44 56 91)(45 57 92)(46 58 93)(47 59 94)(48 60 95)(61 101 128)(62 102 129)(63 103 130)(64 104 131)(65 105 132)(66 106 121)(67 107 122)(68 108 123)(69 97 124)(70 98 125)(71 99 126)(72 100 127)
(1 142 16)(2 143 17)(3 144 18)(4 133 19)(5 134 20)(6 135 21)(7 136 22)(8 137 23)(9 138 24)(10 139 13)(11 140 14)(12 141 15)(25 74 117)(26 75 118)(27 76 119)(28 77 120)(29 78 109)(30 79 110)(31 80 111)(32 81 112)(33 82 113)(34 83 114)(35 84 115)(36 73 116)(37 92 53)(38 93 54)(39 94 55)(40 95 56)(41 96 57)(42 85 58)(43 86 59)(44 87 60)(45 88 49)(46 89 50)(47 90 51)(48 91 52)(61 124 105)(62 125 106)(63 126 107)(64 127 108)(65 128 97)(66 129 98)(67 130 99)(68 131 100)(69 132 101)(70 121 102)(71 122 103)(72 123 104)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 108)(11 97)(12 98)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 61)(23 62)(24 63)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 49)(35 50)(36 51)(37 118)(38 119)(39 120)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 85)(81 86)(82 87)(83 88)(84 89)(121 133)(122 134)(123 135)(124 136)(125 137)(126 138)(127 139)(128 140)(129 141)(130 142)(131 143)(132 144)
(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 90 7 96)(2 89 8 95)(3 88 9 94)(4 87 10 93)(5 86 11 92)(6 85 12 91)(13 38 19 44)(14 37 20 43)(15 48 21 42)(16 47 22 41)(17 46 23 40)(18 45 24 39)(25 123 31 129)(26 122 32 128)(27 121 33 127)(28 132 34 126)(29 131 35 125)(30 130 36 124)(49 138 55 144)(50 137 56 143)(51 136 57 142)(52 135 58 141)(53 134 59 140)(54 133 60 139)(61 110 67 116)(62 109 68 115)(63 120 69 114)(64 119 70 113)(65 118 71 112)(66 117 72 111)(73 105 79 99)(74 104 80 98)(75 103 81 97)(76 102 82 108)(77 101 83 107)(78 100 84 106)
G:=sub<Sym(144)| (1,20,138)(2,21,139)(3,22,140)(4,23,141)(5,24,142)(6,13,143)(7,14,144)(8,15,133)(9,16,134)(10,17,135)(11,18,136)(12,19,137)(25,109,82)(26,110,83)(27,111,84)(28,112,73)(29,113,74)(30,114,75)(31,115,76)(32,116,77)(33,117,78)(34,118,79)(35,119,80)(36,120,81)(37,49,96)(38,50,85)(39,51,86)(40,52,87)(41,53,88)(42,54,89)(43,55,90)(44,56,91)(45,57,92)(46,58,93)(47,59,94)(48,60,95)(61,101,128)(62,102,129)(63,103,130)(64,104,131)(65,105,132)(66,106,121)(67,107,122)(68,108,123)(69,97,124)(70,98,125)(71,99,126)(72,100,127), (1,142,16)(2,143,17)(3,144,18)(4,133,19)(5,134,20)(6,135,21)(7,136,22)(8,137,23)(9,138,24)(10,139,13)(11,140,14)(12,141,15)(25,74,117)(26,75,118)(27,76,119)(28,77,120)(29,78,109)(30,79,110)(31,80,111)(32,81,112)(33,82,113)(34,83,114)(35,84,115)(36,73,116)(37,92,53)(38,93,54)(39,94,55)(40,95,56)(41,96,57)(42,85,58)(43,86,59)(44,87,60)(45,88,49)(46,89,50)(47,90,51)(48,91,52)(61,124,105)(62,125,106)(63,126,107)(64,127,108)(65,128,97)(66,129,98)(67,130,99)(68,131,100)(69,132,101)(70,121,102)(71,122,103)(72,123,104), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,97)(12,98)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,118)(38,119)(39,120)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,85)(81,86)(82,87)(83,88)(84,89)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144), (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), (1,90,7,96)(2,89,8,95)(3,88,9,94)(4,87,10,93)(5,86,11,92)(6,85,12,91)(13,38,19,44)(14,37,20,43)(15,48,21,42)(16,47,22,41)(17,46,23,40)(18,45,24,39)(25,123,31,129)(26,122,32,128)(27,121,33,127)(28,132,34,126)(29,131,35,125)(30,130,36,124)(49,138,55,144)(50,137,56,143)(51,136,57,142)(52,135,58,141)(53,134,59,140)(54,133,60,139)(61,110,67,116)(62,109,68,115)(63,120,69,114)(64,119,70,113)(65,118,71,112)(66,117,72,111)(73,105,79,99)(74,104,80,98)(75,103,81,97)(76,102,82,108)(77,101,83,107)(78,100,84,106)>;
G:=Group( (1,20,138)(2,21,139)(3,22,140)(4,23,141)(5,24,142)(6,13,143)(7,14,144)(8,15,133)(9,16,134)(10,17,135)(11,18,136)(12,19,137)(25,109,82)(26,110,83)(27,111,84)(28,112,73)(29,113,74)(30,114,75)(31,115,76)(32,116,77)(33,117,78)(34,118,79)(35,119,80)(36,120,81)(37,49,96)(38,50,85)(39,51,86)(40,52,87)(41,53,88)(42,54,89)(43,55,90)(44,56,91)(45,57,92)(46,58,93)(47,59,94)(48,60,95)(61,101,128)(62,102,129)(63,103,130)(64,104,131)(65,105,132)(66,106,121)(67,107,122)(68,108,123)(69,97,124)(70,98,125)(71,99,126)(72,100,127), (1,142,16)(2,143,17)(3,144,18)(4,133,19)(5,134,20)(6,135,21)(7,136,22)(8,137,23)(9,138,24)(10,139,13)(11,140,14)(12,141,15)(25,74,117)(26,75,118)(27,76,119)(28,77,120)(29,78,109)(30,79,110)(31,80,111)(32,81,112)(33,82,113)(34,83,114)(35,84,115)(36,73,116)(37,92,53)(38,93,54)(39,94,55)(40,95,56)(41,96,57)(42,85,58)(43,86,59)(44,87,60)(45,88,49)(46,89,50)(47,90,51)(48,91,52)(61,124,105)(62,125,106)(63,126,107)(64,127,108)(65,128,97)(66,129,98)(67,130,99)(68,131,100)(69,132,101)(70,121,102)(71,122,103)(72,123,104), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,108)(11,97)(12,98)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,61)(23,62)(24,63)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,49)(35,50)(36,51)(37,118)(38,119)(39,120)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,85)(81,86)(82,87)(83,88)(84,89)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144), (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), (1,90,7,96)(2,89,8,95)(3,88,9,94)(4,87,10,93)(5,86,11,92)(6,85,12,91)(13,38,19,44)(14,37,20,43)(15,48,21,42)(16,47,22,41)(17,46,23,40)(18,45,24,39)(25,123,31,129)(26,122,32,128)(27,121,33,127)(28,132,34,126)(29,131,35,125)(30,130,36,124)(49,138,55,144)(50,137,56,143)(51,136,57,142)(52,135,58,141)(53,134,59,140)(54,133,60,139)(61,110,67,116)(62,109,68,115)(63,120,69,114)(64,119,70,113)(65,118,71,112)(66,117,72,111)(73,105,79,99)(74,104,80,98)(75,103,81,97)(76,102,82,108)(77,101,83,107)(78,100,84,106) );
G=PermutationGroup([[(1,20,138),(2,21,139),(3,22,140),(4,23,141),(5,24,142),(6,13,143),(7,14,144),(8,15,133),(9,16,134),(10,17,135),(11,18,136),(12,19,137),(25,109,82),(26,110,83),(27,111,84),(28,112,73),(29,113,74),(30,114,75),(31,115,76),(32,116,77),(33,117,78),(34,118,79),(35,119,80),(36,120,81),(37,49,96),(38,50,85),(39,51,86),(40,52,87),(41,53,88),(42,54,89),(43,55,90),(44,56,91),(45,57,92),(46,58,93),(47,59,94),(48,60,95),(61,101,128),(62,102,129),(63,103,130),(64,104,131),(65,105,132),(66,106,121),(67,107,122),(68,108,123),(69,97,124),(70,98,125),(71,99,126),(72,100,127)], [(1,142,16),(2,143,17),(3,144,18),(4,133,19),(5,134,20),(6,135,21),(7,136,22),(8,137,23),(9,138,24),(10,139,13),(11,140,14),(12,141,15),(25,74,117),(26,75,118),(27,76,119),(28,77,120),(29,78,109),(30,79,110),(31,80,111),(32,81,112),(33,82,113),(34,83,114),(35,84,115),(36,73,116),(37,92,53),(38,93,54),(39,94,55),(40,95,56),(41,96,57),(42,85,58),(43,86,59),(44,87,60),(45,88,49),(46,89,50),(47,90,51),(48,91,52),(61,124,105),(62,125,106),(63,126,107),(64,127,108),(65,128,97),(66,129,98),(67,130,99),(68,131,100),(69,132,101),(70,121,102),(71,122,103),(72,123,104)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,105),(8,106),(9,107),(10,108),(11,97),(12,98),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,61),(23,62),(24,63),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,49),(35,50),(36,51),(37,118),(38,119),(39,120),(40,109),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,85),(81,86),(82,87),(83,88),(84,89),(121,133),(122,134),(123,135),(124,136),(125,137),(126,138),(127,139),(128,140),(129,141),(130,142),(131,143),(132,144)], [(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)], [(1,90,7,96),(2,89,8,95),(3,88,9,94),(4,87,10,93),(5,86,11,92),(6,85,12,91),(13,38,19,44),(14,37,20,43),(15,48,21,42),(16,47,22,41),(17,46,23,40),(18,45,24,39),(25,123,31,129),(26,122,32,128),(27,121,33,127),(28,132,34,126),(29,131,35,125),(30,130,36,124),(49,138,55,144),(50,137,56,143),(51,136,57,142),(52,135,58,141),(53,134,59,140),(54,133,60,139),(61,110,67,116),(62,109,68,115),(63,120,69,114),(64,119,70,113),(65,118,71,112),(66,117,72,111),(73,105,79,99),(74,104,80,98),(75,103,81,97),(76,102,82,108),(77,101,83,107),(78,100,84,106)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 12A | 12B | 12C | ··· | 12N | 12O | ··· | 12V | 12W | 12X |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 9 | 9 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 6 | 6 | 18 | 54 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 18 | 18 | 2 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 18 | 18 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D6 | D6 | Dic6 | S32 | S3×Q8 | C2×S32 | S3×Dic6 |
| kernel | C3⋊S3×Dic6 | Dic3×C3⋊S3 | C33⋊4Q8 | C32×Dic6 | C12×C3⋊S3 | C33⋊8Q8 | C3×Dic6 | C4×C3⋊S3 | C3×C3⋊S3 | C3×Dic3 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | C3⋊S3 | C12 | C32 | C6 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 2 | 8 | 1 | 5 | 1 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C3⋊S3×Dic6 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,10,11,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,6,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,1,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[6,9,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C3⋊S3×Dic6 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times {\rm Dic}_6 % in TeX
G:=Group("C3:S3xDic6"); // GroupNames label
G:=SmallGroup(432,663);
// by ID
G=gap.SmallGroup(432,663);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=d^6,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations