direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic9, C9⋊C42, C36⋊2C4, C12.6Dic3, C22.3D18, C6.7(C4×S3), (C2×C4).6D9, C2.2(C4×D9), C3.(C4×Dic3), (C2×C36).6C2, C18.3(C2×C4), (C2×C6).19D6, (C2×C12).14S3, C2.2(C2×Dic9), C6.8(C2×Dic3), (C2×C18).3C22, (C2×Dic9).4C2, SmallGroup(144,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C4×Dic9 |
Generators and relations for C4×Dic9
G = < a,b,c | a4=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×Dic9
►Regular action on 144 points
Generators in S144
(1 92 59 79)(2 93 60 80)(3 94 61 81)(4 95 62 82)(5 96 63 83)(6 97 64 84)(7 98 65 85)(8 99 66 86)(9 100 67 87)(10 101 68 88)(11 102 69 89)(12 103 70 90)(13 104 71 73)(14 105 72 74)(15 106 55 75)(16 107 56 76)(17 108 57 77)(18 91 58 78)(19 49 115 130)(20 50 116 131)(21 51 117 132)(22 52 118 133)(23 53 119 134)(24 54 120 135)(25 37 121 136)(26 38 122 137)(27 39 123 138)(28 40 124 139)(29 41 125 140)(30 42 126 141)(31 43 109 142)(32 44 110 143)(33 45 111 144)(34 46 112 127)(35 47 113 128)(36 48 114 129)
(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 40 10 49)(2 39 11 48)(3 38 12 47)(4 37 13 46)(5 54 14 45)(6 53 15 44)(7 52 16 43)(8 51 17 42)(9 50 18 41)(19 79 28 88)(20 78 29 87)(21 77 30 86)(22 76 31 85)(23 75 32 84)(24 74 33 83)(25 73 34 82)(26 90 35 81)(27 89 36 80)(55 143 64 134)(56 142 65 133)(57 141 66 132)(58 140 67 131)(59 139 68 130)(60 138 69 129)(61 137 70 128)(62 136 71 127)(63 135 72 144)(91 125 100 116)(92 124 101 115)(93 123 102 114)(94 122 103 113)(95 121 104 112)(96 120 105 111)(97 119 106 110)(98 118 107 109)(99 117 108 126)
G:=sub<Sym(144)| (1,92,59,79)(2,93,60,80)(3,94,61,81)(4,95,62,82)(5,96,63,83)(6,97,64,84)(7,98,65,85)(8,99,66,86)(9,100,67,87)(10,101,68,88)(11,102,69,89)(12,103,70,90)(13,104,71,73)(14,105,72,74)(15,106,55,75)(16,107,56,76)(17,108,57,77)(18,91,58,78)(19,49,115,130)(20,50,116,131)(21,51,117,132)(22,52,118,133)(23,53,119,134)(24,54,120,135)(25,37,121,136)(26,38,122,137)(27,39,123,138)(28,40,124,139)(29,41,125,140)(30,42,126,141)(31,43,109,142)(32,44,110,143)(33,45,111,144)(34,46,112,127)(35,47,113,128)(36,48,114,129), (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), (1,40,10,49)(2,39,11,48)(3,38,12,47)(4,37,13,46)(5,54,14,45)(6,53,15,44)(7,52,16,43)(8,51,17,42)(9,50,18,41)(19,79,28,88)(20,78,29,87)(21,77,30,86)(22,76,31,85)(23,75,32,84)(24,74,33,83)(25,73,34,82)(26,90,35,81)(27,89,36,80)(55,143,64,134)(56,142,65,133)(57,141,66,132)(58,140,67,131)(59,139,68,130)(60,138,69,129)(61,137,70,128)(62,136,71,127)(63,135,72,144)(91,125,100,116)(92,124,101,115)(93,123,102,114)(94,122,103,113)(95,121,104,112)(96,120,105,111)(97,119,106,110)(98,118,107,109)(99,117,108,126)>;
G:=Group( (1,92,59,79)(2,93,60,80)(3,94,61,81)(4,95,62,82)(5,96,63,83)(6,97,64,84)(7,98,65,85)(8,99,66,86)(9,100,67,87)(10,101,68,88)(11,102,69,89)(12,103,70,90)(13,104,71,73)(14,105,72,74)(15,106,55,75)(16,107,56,76)(17,108,57,77)(18,91,58,78)(19,49,115,130)(20,50,116,131)(21,51,117,132)(22,52,118,133)(23,53,119,134)(24,54,120,135)(25,37,121,136)(26,38,122,137)(27,39,123,138)(28,40,124,139)(29,41,125,140)(30,42,126,141)(31,43,109,142)(32,44,110,143)(33,45,111,144)(34,46,112,127)(35,47,113,128)(36,48,114,129), (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), (1,40,10,49)(2,39,11,48)(3,38,12,47)(4,37,13,46)(5,54,14,45)(6,53,15,44)(7,52,16,43)(8,51,17,42)(9,50,18,41)(19,79,28,88)(20,78,29,87)(21,77,30,86)(22,76,31,85)(23,75,32,84)(24,74,33,83)(25,73,34,82)(26,90,35,81)(27,89,36,80)(55,143,64,134)(56,142,65,133)(57,141,66,132)(58,140,67,131)(59,139,68,130)(60,138,69,129)(61,137,70,128)(62,136,71,127)(63,135,72,144)(91,125,100,116)(92,124,101,115)(93,123,102,114)(94,122,103,113)(95,121,104,112)(96,120,105,111)(97,119,106,110)(98,118,107,109)(99,117,108,126) );
G=PermutationGroup([[(1,92,59,79),(2,93,60,80),(3,94,61,81),(4,95,62,82),(5,96,63,83),(6,97,64,84),(7,98,65,85),(8,99,66,86),(9,100,67,87),(10,101,68,88),(11,102,69,89),(12,103,70,90),(13,104,71,73),(14,105,72,74),(15,106,55,75),(16,107,56,76),(17,108,57,77),(18,91,58,78),(19,49,115,130),(20,50,116,131),(21,51,117,132),(22,52,118,133),(23,53,119,134),(24,54,120,135),(25,37,121,136),(26,38,122,137),(27,39,123,138),(28,40,124,139),(29,41,125,140),(30,42,126,141),(31,43,109,142),(32,44,110,143),(33,45,111,144),(34,46,112,127),(35,47,113,128),(36,48,114,129)], [(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)], [(1,40,10,49),(2,39,11,48),(3,38,12,47),(4,37,13,46),(5,54,14,45),(6,53,15,44),(7,52,16,43),(8,51,17,42),(9,50,18,41),(19,79,28,88),(20,78,29,87),(21,77,30,86),(22,76,31,85),(23,75,32,84),(24,74,33,83),(25,73,34,82),(26,90,35,81),(27,89,36,80),(55,143,64,134),(56,142,65,133),(57,141,66,132),(58,140,67,131),(59,139,68,130),(60,138,69,129),(61,137,70,128),(62,136,71,127),(63,135,72,144),(91,125,100,116),(92,124,101,115),(93,123,102,114),(94,122,103,113),(95,121,104,112),(96,120,105,111),(97,119,106,110),(98,118,107,109),(99,117,108,126)]])
C4×Dic9 is a maximal subgroup of
Dic9⋊C8 C72⋊C4 Dic18⋊C4 Q8⋊3Dic9 C42×D9 C42⋊2D9 C23.16D18 C23.8D18 Dic9⋊4D4 Dic9.D4 Dic9⋊3Q8 C36⋊Q8 Dic9.Q8 C36.3Q8 C4⋊C4⋊7D9 D36⋊C4 C4⋊C4⋊D9 C23.26D18 C36.17D4 C36⋊D4 Dic9⋊Q8 C36.23D4
C4×Dic9 is a maximal quotient of
C42.D9 C72⋊C4 C18.C42
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | D9 | C4×S3 | Dic9 | D18 | C4×D9 |
| kernel | C4×Dic9 | C2×Dic9 | C2×C36 | Dic9 | C36 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 1 | 3 | 4 | 6 | 3 | 12 |
Matrix representation of C4×Dic9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 20 |
| 0 | 0 | 17 | 11 |
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 32 |
| 0 | 0 | 27 | 32 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,6,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,1,0,0,0,0,31,17,0,0,20,11],[6,0,0,0,0,1,0,0,0,0,5,27,0,0,32,32] >;
C4×Dic9 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_9 % in TeX
G:=Group("C4xDic9"); // GroupNames label
G:=SmallGroup(144,11);
// by ID
G=gap.SmallGroup(144,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,55,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^4=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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