direct product, metabelian, supersoluble, monomial
Aliases: S3×Dic18, D6.8D18, C36.34D6, C12.23D18, Dic9.2D6, Dic3.7D18, (S3×C9)⋊Q8, C9⋊2(S3×Q8), C12.48S32, (C4×S3).1D9, C4.12(S3×D9), (S3×Dic9).C2, (C3×S3).Dic6, (S3×C36).1C2, (S3×C12).4S3, (S3×C6).25D6, C3⋊1(C2×Dic18), (C3×C12).93D6, C3.2(S3×Dic6), C12.D9⋊3C2, C6.5(C22×D9), (C3×Dic18)⋊3C2, C9⋊Dic6⋊1C2, C18.5(C22×S3), (C3×C36).5C22, (C3×C18).5C23, (S3×C18).8C22, (C3×Dic3).28D6, C9⋊Dic3.2C22, C32.2(C2×Dic6), (C3×Dic9).2C22, (C9×Dic3).7C22, (C3×C9)⋊3(C2×Q8), C2.9(C2×S3×D9), C6.24(C2×S32), (C3×C6).73(C22×S3), SmallGroup(432,284)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Dic18
G = < a,b,c,d | a3=b2=c36=1, d2=c18, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 656 in 126 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×C9, Dic9, Dic9, C36, C36, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×Dic6, S3×Q8, S3×C9, C3×C18, Dic18, Dic18, C2×Dic9, C2×C36, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, S3×C18, C2×Dic18, S3×Dic6, C9⋊Dic6, S3×Dic9, C3×Dic18, S3×C36, C12.D9, S3×Dic18
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, S32, C2×Dic6, S3×Q8, Dic18, C22×D9, C2×S32, S3×D9, C2×Dic18, S3×Dic6, C2×S3×D9, S3×Dic18
►On 144 points
Generators in S144
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)(73 85 97)(74 86 98)(75 87 99)(76 88 100)(77 89 101)(78 90 102)(79 91 103)(80 92 104)(81 93 105)(82 94 106)(83 95 107)(84 96 108)(109 133 121)(110 134 122)(111 135 123)(112 136 124)(113 137 125)(114 138 126)(115 139 127)(116 140 128)(117 141 129)(118 142 130)(119 143 131)(120 144 132)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)
(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 41 19 59)(2 40 20 58)(3 39 21 57)(4 38 22 56)(5 37 23 55)(6 72 24 54)(7 71 25 53)(8 70 26 52)(9 69 27 51)(10 68 28 50)(11 67 29 49)(12 66 30 48)(13 65 31 47)(14 64 32 46)(15 63 33 45)(16 62 34 44)(17 61 35 43)(18 60 36 42)(73 140 91 122)(74 139 92 121)(75 138 93 120)(76 137 94 119)(77 136 95 118)(78 135 96 117)(79 134 97 116)(80 133 98 115)(81 132 99 114)(82 131 100 113)(83 130 101 112)(84 129 102 111)(85 128 103 110)(86 127 104 109)(87 126 105 144)(88 125 106 143)(89 124 107 142)(90 123 108 141)
G:=sub<Sym(144)| (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,85,97)(74,86,98)(75,87,99)(76,88,100)(77,89,101)(78,90,102)(79,91,103)(80,92,104)(81,93,105)(82,94,106)(83,95,107)(84,96,108)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135), (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), (1,41,19,59)(2,40,20,58)(3,39,21,57)(4,38,22,56)(5,37,23,55)(6,72,24,54)(7,71,25,53)(8,70,26,52)(9,69,27,51)(10,68,28,50)(11,67,29,49)(12,66,30,48)(13,65,31,47)(14,64,32,46)(15,63,33,45)(16,62,34,44)(17,61,35,43)(18,60,36,42)(73,140,91,122)(74,139,92,121)(75,138,93,120)(76,137,94,119)(77,136,95,118)(78,135,96,117)(79,134,97,116)(80,133,98,115)(81,132,99,114)(82,131,100,113)(83,130,101,112)(84,129,102,111)(85,128,103,110)(86,127,104,109)(87,126,105,144)(88,125,106,143)(89,124,107,142)(90,123,108,141)>;
G:=Group( (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,85,97)(74,86,98)(75,87,99)(76,88,100)(77,89,101)(78,90,102)(79,91,103)(80,92,104)(81,93,105)(82,94,106)(83,95,107)(84,96,108)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,136)(38,137)(39,138)(40,139)(41,140)(42,141)(43,142)(44,143)(45,144)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135), (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), (1,41,19,59)(2,40,20,58)(3,39,21,57)(4,38,22,56)(5,37,23,55)(6,72,24,54)(7,71,25,53)(8,70,26,52)(9,69,27,51)(10,68,28,50)(11,67,29,49)(12,66,30,48)(13,65,31,47)(14,64,32,46)(15,63,33,45)(16,62,34,44)(17,61,35,43)(18,60,36,42)(73,140,91,122)(74,139,92,121)(75,138,93,120)(76,137,94,119)(77,136,95,118)(78,135,96,117)(79,134,97,116)(80,133,98,115)(81,132,99,114)(82,131,100,113)(83,130,101,112)(84,129,102,111)(85,128,103,110)(86,127,104,109)(87,126,105,144)(88,125,106,143)(89,124,107,142)(90,123,108,141) );
G=PermutationGroup([[(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,49,61),(38,50,62),(39,51,63),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69),(46,58,70),(47,59,71),(48,60,72),(73,85,97),(74,86,98),(75,87,99),(76,88,100),(77,89,101),(78,90,102),(79,91,103),(80,92,104),(81,93,105),(82,94,106),(83,95,107),(84,96,108),(109,133,121),(110,134,122),(111,135,123),(112,136,124),(113,137,125),(114,138,126),(115,139,127),(116,140,128),(117,141,129),(118,142,130),(119,143,131),(120,144,132)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,136),(38,137),(39,138),(40,139),(41,140),(42,141),(43,142),(44,143),(45,144),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135)], [(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)], [(1,41,19,59),(2,40,20,58),(3,39,21,57),(4,38,22,56),(5,37,23,55),(6,72,24,54),(7,71,25,53),(8,70,26,52),(9,69,27,51),(10,68,28,50),(11,67,29,49),(12,66,30,48),(13,65,31,47),(14,64,32,46),(15,63,33,45),(16,62,34,44),(17,61,35,43),(18,60,36,42),(73,140,91,122),(74,139,92,121),(75,138,93,120),(76,137,94,119),(77,136,95,118),(78,135,96,117),(79,134,97,116),(80,133,98,115),(81,132,99,114),(82,131,100,113),(83,130,101,112),(84,129,102,111),(85,128,103,110),(86,127,104,109),(87,126,105,144),(88,125,106,143),(89,124,107,142),(90,123,108,141)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36L | 36M | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 2 | 6 | 18 | 18 | 54 | 54 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | + | + | - | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D6 | D6 | D6 | D9 | Dic6 | D18 | D18 | D18 | Dic18 | S32 | S3×Q8 | C2×S32 | S3×D9 | S3×Dic6 | C2×S3×D9 | S3×Dic18 |
| kernel | S3×Dic18 | C9⋊Dic6 | S3×Dic9 | C3×Dic18 | S3×C36 | C12.D9 | Dic18 | S3×C12 | S3×C9 | Dic9 | C36 | C3×Dic3 | C3×C12 | S3×C6 | C4×S3 | C3×S3 | Dic3 | C12 | D6 | S3 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 12 | 1 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of S3×Dic18 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 34 | 0 | 0 | 0 | 0 |
| 26 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 27 | 4 | 0 | 0 | 0 | 0 |
| 21 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 34 | 0 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,26,0,0,0,0,34,22,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,7,0,0,0,0,0,0,16],[27,21,0,0,0,0,4,10,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,34,0,0,0,0,12,0] >;
S3×Dic18 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_{18} % in TeX
G:=Group("S3xDic18"); // GroupNames label
G:=SmallGroup(432,284);
// by ID
G=gap.SmallGroup(432,284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^36=1,d^2=c^18,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations