direct product, metabelian, supersoluble, monomial
Aliases: C9×Dic3⋊C4, Dic3⋊C36, C18.9Dic6, C6.5(D4×C9), C6.1(Q8×C9), C2.4(S3×C36), (C6×C36).4C2, C6.3(C2×C36), (C2×C36).3S3, (C3×C18).4Q8, C6.36(S3×C12), C18.24(C4×S3), (C2×C12).2C18, (C6×C12).21C6, (C9×Dic3)⋊3C4, (C2×C18).47D6, (C3×C18).33D4, C2.1(C9×Dic6), C62.49(C2×C6), C22.4(S3×C18), (C6×Dic3).1C6, C6.14(C3×Dic6), C18.29(C3⋊D4), (C6×C18).22C22, (C3×Dic3).4C12, (Dic3×C18).3C2, (C2×Dic3).1C18, C3⋊1(C9×C4⋊C4), (C3×C9)⋊4(C4⋊C4), (C2×C4).1(S3×C9), C2.1(C9×C3⋊D4), (C3×C6).9(C3×Q8), (C2×C6).81(S3×C6), (C2×C12).2(C3×S3), (C2×C6).7(C2×C18), (C3×C6).53(C3×D4), (C3×Dic3⋊C4).C3, C6.43(C3×C3⋊D4), C32.2(C3×C4⋊C4), (C3×C6).40(C2×C12), (C3×C18).16(C2×C4), C3.4(C3×Dic3⋊C4), SmallGroup(432,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×Dic3⋊C4
G = < a,b,c,d | a9=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >
Subgroups: 164 in 94 conjugacy classes, 51 normal (45 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3×C12, C62, Dic3⋊C4, C3×C4⋊C4, C3×C18, C2×C36, C2×C36, C6×Dic3, C6×C12, C9×Dic3, C9×Dic3, C3×C36, C6×C18, C9×C4⋊C4, C3×Dic3⋊C4, Dic3×C18, C6×C36, C9×Dic3⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C9, C12, D6, C2×C6, C4⋊C4, C18, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C36, C2×C18, S3×C6, Dic3⋊C4, C3×C4⋊C4, S3×C9, C2×C36, D4×C9, Q8×C9, C3×Dic6, S3×C12, C3×C3⋊D4, S3×C18, C9×C4⋊C4, C3×Dic3⋊C4, C9×Dic6, S3×C36, C9×C3⋊D4, C9×Dic3⋊C4
►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 41 7 38 4 44)(2 42 8 39 5 45)(3 43 9 40 6 37)(10 136 13 139 16 142)(11 137 14 140 17 143)(12 138 15 141 18 144)(19 129 22 132 25 135)(20 130 23 133 26 127)(21 131 24 134 27 128)(28 53 34 50 31 47)(29 54 35 51 32 48)(30 46 36 52 33 49)(55 84 61 90 58 87)(56 85 62 82 59 88)(57 86 63 83 60 89)(64 80 70 77 67 74)(65 81 71 78 68 75)(66 73 72 79 69 76)(91 123 94 126 97 120)(92 124 95 118 98 121)(93 125 96 119 99 122)(100 110 103 113 106 116)(101 111 104 114 107 117)(102 112 105 115 108 109)
(1 125 38 99)(2 126 39 91)(3 118 40 92)(4 119 41 93)(5 120 42 94)(6 121 43 95)(7 122 44 96)(8 123 45 97)(9 124 37 98)(10 63 139 89)(11 55 140 90)(12 56 141 82)(13 57 142 83)(14 58 143 84)(15 59 144 85)(16 60 136 86)(17 61 137 87)(18 62 138 88)(19 64 132 77)(20 65 133 78)(21 66 134 79)(22 67 135 80)(23 68 127 81)(24 69 128 73)(25 70 129 74)(26 71 130 75)(27 72 131 76)(28 100 50 113)(29 101 51 114)(30 102 52 115)(31 103 53 116)(32 104 54 117)(33 105 46 109)(34 106 47 110)(35 107 48 111)(36 108 49 112)
(1 63 31 80)(2 55 32 81)(3 56 33 73)(4 57 34 74)(5 58 35 75)(6 59 36 76)(7 60 28 77)(8 61 29 78)(9 62 30 79)(10 103 135 125)(11 104 127 126)(12 105 128 118)(13 106 129 119)(14 107 130 120)(15 108 131 121)(16 100 132 122)(17 101 133 123)(18 102 134 124)(19 96 136 113)(20 97 137 114)(21 98 138 115)(22 99 139 116)(23 91 140 117)(24 92 141 109)(25 93 142 110)(26 94 143 111)(27 95 144 112)(37 88 52 66)(38 89 53 67)(39 90 54 68)(40 82 46 69)(41 83 47 70)(42 84 48 71)(43 85 49 72)(44 86 50 64)(45 87 51 65)
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), (1,41,7,38,4,44)(2,42,8,39,5,45)(3,43,9,40,6,37)(10,136,13,139,16,142)(11,137,14,140,17,143)(12,138,15,141,18,144)(19,129,22,132,25,135)(20,130,23,133,26,127)(21,131,24,134,27,128)(28,53,34,50,31,47)(29,54,35,51,32,48)(30,46,36,52,33,49)(55,84,61,90,58,87)(56,85,62,82,59,88)(57,86,63,83,60,89)(64,80,70,77,67,74)(65,81,71,78,68,75)(66,73,72,79,69,76)(91,123,94,126,97,120)(92,124,95,118,98,121)(93,125,96,119,99,122)(100,110,103,113,106,116)(101,111,104,114,107,117)(102,112,105,115,108,109), (1,125,38,99)(2,126,39,91)(3,118,40,92)(4,119,41,93)(5,120,42,94)(6,121,43,95)(7,122,44,96)(8,123,45,97)(9,124,37,98)(10,63,139,89)(11,55,140,90)(12,56,141,82)(13,57,142,83)(14,58,143,84)(15,59,144,85)(16,60,136,86)(17,61,137,87)(18,62,138,88)(19,64,132,77)(20,65,133,78)(21,66,134,79)(22,67,135,80)(23,68,127,81)(24,69,128,73)(25,70,129,74)(26,71,130,75)(27,72,131,76)(28,100,50,113)(29,101,51,114)(30,102,52,115)(31,103,53,116)(32,104,54,117)(33,105,46,109)(34,106,47,110)(35,107,48,111)(36,108,49,112), (1,63,31,80)(2,55,32,81)(3,56,33,73)(4,57,34,74)(5,58,35,75)(6,59,36,76)(7,60,28,77)(8,61,29,78)(9,62,30,79)(10,103,135,125)(11,104,127,126)(12,105,128,118)(13,106,129,119)(14,107,130,120)(15,108,131,121)(16,100,132,122)(17,101,133,123)(18,102,134,124)(19,96,136,113)(20,97,137,114)(21,98,138,115)(22,99,139,116)(23,91,140,117)(24,92,141,109)(25,93,142,110)(26,94,143,111)(27,95,144,112)(37,88,52,66)(38,89,53,67)(39,90,54,68)(40,82,46,69)(41,83,47,70)(42,84,48,71)(43,85,49,72)(44,86,50,64)(45,87,51,65)>;
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), (1,41,7,38,4,44)(2,42,8,39,5,45)(3,43,9,40,6,37)(10,136,13,139,16,142)(11,137,14,140,17,143)(12,138,15,141,18,144)(19,129,22,132,25,135)(20,130,23,133,26,127)(21,131,24,134,27,128)(28,53,34,50,31,47)(29,54,35,51,32,48)(30,46,36,52,33,49)(55,84,61,90,58,87)(56,85,62,82,59,88)(57,86,63,83,60,89)(64,80,70,77,67,74)(65,81,71,78,68,75)(66,73,72,79,69,76)(91,123,94,126,97,120)(92,124,95,118,98,121)(93,125,96,119,99,122)(100,110,103,113,106,116)(101,111,104,114,107,117)(102,112,105,115,108,109), (1,125,38,99)(2,126,39,91)(3,118,40,92)(4,119,41,93)(5,120,42,94)(6,121,43,95)(7,122,44,96)(8,123,45,97)(9,124,37,98)(10,63,139,89)(11,55,140,90)(12,56,141,82)(13,57,142,83)(14,58,143,84)(15,59,144,85)(16,60,136,86)(17,61,137,87)(18,62,138,88)(19,64,132,77)(20,65,133,78)(21,66,134,79)(22,67,135,80)(23,68,127,81)(24,69,128,73)(25,70,129,74)(26,71,130,75)(27,72,131,76)(28,100,50,113)(29,101,51,114)(30,102,52,115)(31,103,53,116)(32,104,54,117)(33,105,46,109)(34,106,47,110)(35,107,48,111)(36,108,49,112), (1,63,31,80)(2,55,32,81)(3,56,33,73)(4,57,34,74)(5,58,35,75)(6,59,36,76)(7,60,28,77)(8,61,29,78)(9,62,30,79)(10,103,135,125)(11,104,127,126)(12,105,128,118)(13,106,129,119)(14,107,130,120)(15,108,131,121)(16,100,132,122)(17,101,133,123)(18,102,134,124)(19,96,136,113)(20,97,137,114)(21,98,138,115)(22,99,139,116)(23,91,140,117)(24,92,141,109)(25,93,142,110)(26,94,143,111)(27,95,144,112)(37,88,52,66)(38,89,53,67)(39,90,54,68)(40,82,46,69)(41,83,47,70)(42,84,48,71)(43,85,49,72)(44,86,50,64)(45,87,51,65) );
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)], [(1,41,7,38,4,44),(2,42,8,39,5,45),(3,43,9,40,6,37),(10,136,13,139,16,142),(11,137,14,140,17,143),(12,138,15,141,18,144),(19,129,22,132,25,135),(20,130,23,133,26,127),(21,131,24,134,27,128),(28,53,34,50,31,47),(29,54,35,51,32,48),(30,46,36,52,33,49),(55,84,61,90,58,87),(56,85,62,82,59,88),(57,86,63,83,60,89),(64,80,70,77,67,74),(65,81,71,78,68,75),(66,73,72,79,69,76),(91,123,94,126,97,120),(92,124,95,118,98,121),(93,125,96,119,99,122),(100,110,103,113,106,116),(101,111,104,114,107,117),(102,112,105,115,108,109)], [(1,125,38,99),(2,126,39,91),(3,118,40,92),(4,119,41,93),(5,120,42,94),(6,121,43,95),(7,122,44,96),(8,123,45,97),(9,124,37,98),(10,63,139,89),(11,55,140,90),(12,56,141,82),(13,57,142,83),(14,58,143,84),(15,59,144,85),(16,60,136,86),(17,61,137,87),(18,62,138,88),(19,64,132,77),(20,65,133,78),(21,66,134,79),(22,67,135,80),(23,68,127,81),(24,69,128,73),(25,70,129,74),(26,71,130,75),(27,72,131,76),(28,100,50,113),(29,101,51,114),(30,102,52,115),(31,103,53,116),(32,104,54,117),(33,105,46,109),(34,106,47,110),(35,107,48,111),(36,108,49,112)], [(1,63,31,80),(2,55,32,81),(3,56,33,73),(4,57,34,74),(5,58,35,75),(6,59,36,76),(7,60,28,77),(8,61,29,78),(9,62,30,79),(10,103,135,125),(11,104,127,126),(12,105,128,118),(13,106,129,119),(14,107,130,120),(15,108,131,121),(16,100,132,122),(17,101,133,123),(18,102,134,124),(19,96,136,113),(20,97,137,114),(21,98,138,115),(22,99,139,116),(23,91,140,117),(24,92,141,109),(25,93,142,110),(26,94,143,111),(27,95,144,112),(37,88,52,66),(38,89,53,67),(39,90,54,68),(40,82,46,69),(41,83,47,70),(42,84,48,71),(43,85,49,72),(44,86,50,64),(45,87,51,65)]])
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 | D6 | C3×S3 | Dic6 | C4×S3 | C3⋊D4 | C3×D4 | C3×Q8 | S3×C6 | S3×C9 | D4×C9 | Q8×C9 | C3×Dic6 | S3×C12 | C3×C3⋊D4 | S3×C18 | C9×Dic6 | S3×C36 | C9×C3⋊D4 |
| kernel | C9×Dic3⋊C4 | Dic3×C18 | C6×C36 | C3×Dic3⋊C4 | C9×Dic3 | C6×Dic3 | C6×C12 | Dic3⋊C4 | C3×Dic3 | C2×Dic3 | C2×C12 | Dic3 | C2×C36 | C3×C18 | C3×C18 | C2×C18 | C2×C12 | C18 | C18 | C18 | C3×C6 | C3×C6 | C2×C6 | C2×C4 | C6 | C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 24 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 4 | 4 | 4 | 6 | 12 | 12 | 12 |
Matrix representation of C9×Dic3⋊C4 ►in GL3(𝔽37) generated by
| 33 | 0 | 0 |
| 0 | 34 | 0 |
| 0 | 0 | 34 |
| 1 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 11 |
| 36 | 0 | 0 |
| 0 | 0 | 36 |
| 0 | 1 | 0 |
| 31 | 0 | 0 |
| 0 | 31 | 0 |
| 0 | 0 | 6 |
G:=sub<GL(3,GF(37))| [33,0,0,0,34,0,0,0,34],[1,0,0,0,27,0,0,0,11],[36,0,0,0,0,1,0,36,0],[31,0,0,0,31,0,0,0,6] >;
C9×Dic3⋊C4 in GAP, Magma, Sage, TeX
C_9\times {\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("C9xDic3:C4"); // GroupNames label
G:=SmallGroup(432,132);
// by ID
G=gap.SmallGroup(432,132);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,365,92,268,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations