direct product, metacyclic, supersoluble, monomial
Aliases: C9×C4⋊Dic3, C12⋊1C36, C36⋊3Dic3, C18.22D12, C18.10Dic6, C4⋊(C9×Dic3), (C3×C36)⋊6C4, C6.4(D4×C9), C6.2(Q8×C9), C6.8(C2×C36), C2.1(C9×D12), (C3×C18).5Q8, (C6×C12).34C6, (C2×C12).3C18, (C2×C36).19S3, (C6×C36).18C2, (C2×C18).48D6, C6.34(C3×D12), (C3×C18).22D4, C2.2(C9×Dic6), (C3×C12).17C12, C62.50(C2×C6), C22.5(S3×C18), (C6×Dic3).2C6, C2.4(Dic3×C18), C6.15(C3×Dic6), C6.32(C6×Dic3), (C6×C18).23C22, (Dic3×C18).4C2, (C2×Dic3).2C18, C12.20(C3×Dic3), C18.20(C2×Dic3), C3⋊2(C9×C4⋊C4), (C3×C9)⋊5(C4⋊C4), (C2×C4).3(S3×C9), (C3×C4⋊Dic3).C3, (C2×C6).8(C2×C18), (C2×C6).82(S3×C6), (C3×C6).42(C3×D4), C32.3(C3×C4⋊C4), C3.4(C3×C4⋊Dic3), (C3×C6).10(C3×Q8), (C3×C18).30(C2×C4), (C2×C12).41(C3×S3), (C3×C6).53(C2×C12), SmallGroup(432,133)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4⋊Dic3
G = < a,b,c,d | a9=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 164 in 94 conjugacy classes, 57 normal (39 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C3×C18, C2×C36, C2×C36, C6×Dic3, C6×C12, C9×Dic3, C3×C36, C6×C18, C9×C4⋊C4, C3×C4⋊Dic3, Dic3×C18, C6×C36, C9×C4⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C4⋊C4, C18, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C36, C2×C18, C3×Dic3, S3×C6, C4⋊Dic3, C3×C4⋊C4, S3×C9, C2×C36, D4×C9, Q8×C9, C3×Dic6, C3×D12, C6×Dic3, C9×Dic3, S3×C18, C9×C4⋊C4, C3×C4⋊Dic3, C9×Dic6, C9×D12, Dic3×C18, C9×C4⋊Dic3
►On 144 points
Generators in S144
(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 67 37 79)(2 68 38 80)(3 69 39 81)(4 70 40 73)(5 71 41 74)(6 72 42 75)(7 64 43 76)(8 65 44 77)(9 66 45 78)(10 104 140 116)(11 105 141 117)(12 106 142 109)(13 107 143 110)(14 108 144 111)(15 100 136 112)(16 101 137 113)(17 102 138 114)(18 103 139 115)(19 96 132 121)(20 97 133 122)(21 98 134 123)(22 99 135 124)(23 91 127 125)(24 92 128 126)(25 93 129 118)(26 94 130 119)(27 95 131 120)(28 87 51 62)(29 88 52 63)(30 89 53 55)(31 90 54 56)(32 82 46 57)(33 83 47 58)(34 84 48 59)(35 85 49 60)(36 86 50 61)
(1 32 7 29 4 35)(2 33 8 30 5 36)(3 34 9 31 6 28)(10 133 13 127 16 130)(11 134 14 128 17 131)(12 135 15 129 18 132)(19 142 22 136 25 139)(20 143 23 137 26 140)(21 144 24 138 27 141)(37 46 43 52 40 49)(38 47 44 53 41 50)(39 48 45 54 42 51)(55 74 61 80 58 77)(56 75 62 81 59 78)(57 76 63 73 60 79)(64 88 70 85 67 82)(65 89 71 86 68 83)(66 90 72 87 69 84)(91 113 94 116 97 110)(92 114 95 117 98 111)(93 115 96 109 99 112)(100 118 103 121 106 124)(101 119 104 122 107 125)(102 120 105 123 108 126)
(1 99 29 115)(2 91 30 116)(3 92 31 117)(4 93 32 109)(5 94 33 110)(6 95 34 111)(7 96 35 112)(8 97 36 113)(9 98 28 114)(10 80 127 55)(11 81 128 56)(12 73 129 57)(13 74 130 58)(14 75 131 59)(15 76 132 60)(16 77 133 61)(17 78 134 62)(18 79 135 63)(19 85 136 64)(20 86 137 65)(21 87 138 66)(22 88 139 67)(23 89 140 68)(24 90 141 69)(25 82 142 70)(26 83 143 71)(27 84 144 72)(37 124 52 103)(38 125 53 104)(39 126 54 105)(40 118 46 106)(41 119 47 107)(42 120 48 108)(43 121 49 100)(44 122 50 101)(45 123 51 102)
G:=sub<Sym(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), (1,67,37,79)(2,68,38,80)(3,69,39,81)(4,70,40,73)(5,71,41,74)(6,72,42,75)(7,64,43,76)(8,65,44,77)(9,66,45,78)(10,104,140,116)(11,105,141,117)(12,106,142,109)(13,107,143,110)(14,108,144,111)(15,100,136,112)(16,101,137,113)(17,102,138,114)(18,103,139,115)(19,96,132,121)(20,97,133,122)(21,98,134,123)(22,99,135,124)(23,91,127,125)(24,92,128,126)(25,93,129,118)(26,94,130,119)(27,95,131,120)(28,87,51,62)(29,88,52,63)(30,89,53,55)(31,90,54,56)(32,82,46,57)(33,83,47,58)(34,84,48,59)(35,85,49,60)(36,86,50,61), (1,32,7,29,4,35)(2,33,8,30,5,36)(3,34,9,31,6,28)(10,133,13,127,16,130)(11,134,14,128,17,131)(12,135,15,129,18,132)(19,142,22,136,25,139)(20,143,23,137,26,140)(21,144,24,138,27,141)(37,46,43,52,40,49)(38,47,44,53,41,50)(39,48,45,54,42,51)(55,74,61,80,58,77)(56,75,62,81,59,78)(57,76,63,73,60,79)(64,88,70,85,67,82)(65,89,71,86,68,83)(66,90,72,87,69,84)(91,113,94,116,97,110)(92,114,95,117,98,111)(93,115,96,109,99,112)(100,118,103,121,106,124)(101,119,104,122,107,125)(102,120,105,123,108,126), (1,99,29,115)(2,91,30,116)(3,92,31,117)(4,93,32,109)(5,94,33,110)(6,95,34,111)(7,96,35,112)(8,97,36,113)(9,98,28,114)(10,80,127,55)(11,81,128,56)(12,73,129,57)(13,74,130,58)(14,75,131,59)(15,76,132,60)(16,77,133,61)(17,78,134,62)(18,79,135,63)(19,85,136,64)(20,86,137,65)(21,87,138,66)(22,88,139,67)(23,89,140,68)(24,90,141,69)(25,82,142,70)(26,83,143,71)(27,84,144,72)(37,124,52,103)(38,125,53,104)(39,126,54,105)(40,118,46,106)(41,119,47,107)(42,120,48,108)(43,121,49,100)(44,122,50,101)(45,123,51,102)>;
G:=Group( (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), (1,67,37,79)(2,68,38,80)(3,69,39,81)(4,70,40,73)(5,71,41,74)(6,72,42,75)(7,64,43,76)(8,65,44,77)(9,66,45,78)(10,104,140,116)(11,105,141,117)(12,106,142,109)(13,107,143,110)(14,108,144,111)(15,100,136,112)(16,101,137,113)(17,102,138,114)(18,103,139,115)(19,96,132,121)(20,97,133,122)(21,98,134,123)(22,99,135,124)(23,91,127,125)(24,92,128,126)(25,93,129,118)(26,94,130,119)(27,95,131,120)(28,87,51,62)(29,88,52,63)(30,89,53,55)(31,90,54,56)(32,82,46,57)(33,83,47,58)(34,84,48,59)(35,85,49,60)(36,86,50,61), (1,32,7,29,4,35)(2,33,8,30,5,36)(3,34,9,31,6,28)(10,133,13,127,16,130)(11,134,14,128,17,131)(12,135,15,129,18,132)(19,142,22,136,25,139)(20,143,23,137,26,140)(21,144,24,138,27,141)(37,46,43,52,40,49)(38,47,44,53,41,50)(39,48,45,54,42,51)(55,74,61,80,58,77)(56,75,62,81,59,78)(57,76,63,73,60,79)(64,88,70,85,67,82)(65,89,71,86,68,83)(66,90,72,87,69,84)(91,113,94,116,97,110)(92,114,95,117,98,111)(93,115,96,109,99,112)(100,118,103,121,106,124)(101,119,104,122,107,125)(102,120,105,123,108,126), (1,99,29,115)(2,91,30,116)(3,92,31,117)(4,93,32,109)(5,94,33,110)(6,95,34,111)(7,96,35,112)(8,97,36,113)(9,98,28,114)(10,80,127,55)(11,81,128,56)(12,73,129,57)(13,74,130,58)(14,75,131,59)(15,76,132,60)(16,77,133,61)(17,78,134,62)(18,79,135,63)(19,85,136,64)(20,86,137,65)(21,87,138,66)(22,88,139,67)(23,89,140,68)(24,90,141,69)(25,82,142,70)(26,83,143,71)(27,84,144,72)(37,124,52,103)(38,125,53,104)(39,126,54,105)(40,118,46,106)(41,119,47,107)(42,120,48,108)(43,121,49,100)(44,122,50,101)(45,123,51,102) );
G=PermutationGroup([[(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)], [(1,67,37,79),(2,68,38,80),(3,69,39,81),(4,70,40,73),(5,71,41,74),(6,72,42,75),(7,64,43,76),(8,65,44,77),(9,66,45,78),(10,104,140,116),(11,105,141,117),(12,106,142,109),(13,107,143,110),(14,108,144,111),(15,100,136,112),(16,101,137,113),(17,102,138,114),(18,103,139,115),(19,96,132,121),(20,97,133,122),(21,98,134,123),(22,99,135,124),(23,91,127,125),(24,92,128,126),(25,93,129,118),(26,94,130,119),(27,95,131,120),(28,87,51,62),(29,88,52,63),(30,89,53,55),(31,90,54,56),(32,82,46,57),(33,83,47,58),(34,84,48,59),(35,85,49,60),(36,86,50,61)], [(1,32,7,29,4,35),(2,33,8,30,5,36),(3,34,9,31,6,28),(10,133,13,127,16,130),(11,134,14,128,17,131),(12,135,15,129,18,132),(19,142,22,136,25,139),(20,143,23,137,26,140),(21,144,24,138,27,141),(37,46,43,52,40,49),(38,47,44,53,41,50),(39,48,45,54,42,51),(55,74,61,80,58,77),(56,75,62,81,59,78),(57,76,63,73,60,79),(64,88,70,85,67,82),(65,89,71,86,68,83),(66,90,72,87,69,84),(91,113,94,116,97,110),(92,114,95,117,98,111),(93,115,96,109,99,112),(100,118,103,121,106,124),(101,119,104,122,107,125),(102,120,105,123,108,126)], [(1,99,29,115),(2,91,30,116),(3,92,31,117),(4,93,32,109),(5,94,33,110),(6,95,34,111),(7,96,35,112),(8,97,36,113),(9,98,28,114),(10,80,127,55),(11,81,128,56),(12,73,129,57),(13,74,130,58),(14,75,131,59),(15,76,132,60),(16,77,133,61),(17,78,134,62),(18,79,135,63),(19,85,136,64),(20,86,137,65),(21,87,138,66),(22,88,139,67),(23,89,140,68),(24,90,141,69),(25,82,142,70),(26,83,143,71),(27,84,144,72),(37,124,52,103),(38,125,53,104),(39,126,54,105),(40,118,46,106),(41,119,47,107),(42,120,48,108),(43,121,49,100),(44,122,50,101),(45,123,51,102)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12P | 12Q | ··· | 12X | 18A | ··· | 18R | 18S | ··· | 18AJ | 36A | ··· | 36AJ | 36AK | ··· | 36BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | Dic6 | D12 | C3×D4 | C3×Q8 | C3×Dic3 | S3×C6 | S3×C9 | D4×C9 | Q8×C9 | C3×Dic6 | C3×D12 | C9×Dic3 | S3×C18 | C9×Dic6 | C9×D12 |
| kernel | C9×C4⋊Dic3 | Dic3×C18 | C6×C36 | C3×C4⋊Dic3 | C3×C36 | C6×Dic3 | C6×C12 | C4⋊Dic3 | C3×C12 | C2×Dic3 | C2×C12 | C12 | C2×C36 | C3×C18 | C3×C18 | C36 | C2×C18 | C2×C12 | C18 | C18 | C3×C6 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 24 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 6 | 6 | 6 | 4 | 4 | 12 | 6 | 12 | 12 |
Matrix representation of C9×C4⋊Dic3 ►in GL3(𝔽37) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 36 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 31 |
| 36 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 27 |
| 6 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 36 | 0 |
G:=sub<GL(3,GF(37))| [12,0,0,0,12,0,0,0,12],[36,0,0,0,6,0,0,0,31],[36,0,0,0,11,0,0,0,27],[6,0,0,0,0,36,0,1,0] >;
C9×C4⋊Dic3 in GAP, Magma, Sage, TeX
C_9\times C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C9xC4:Dic3"); // GroupNames label
G:=SmallGroup(432,133);
// by ID
G=gap.SmallGroup(432,133);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,268,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=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