metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic9, C36⋊1C4, C2.1D36, C6.4D12, C18.4D4, C18.2Q8, C6.4Dic6, C2.2Dic18, C12.2Dic3, C22.5D18, C9⋊2(C4⋊C4), (C2×C4).3D9, (C2×C12).5S3, C18.8(C2×C4), (C2×C36).2C2, (C2×C6).21D6, C3.(C4⋊Dic3), C2.4(C2×Dic9), C6.9(C2×Dic3), (C2×C18).5C22, (C2×Dic9).2C2, SmallGroup(144,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic9
G = < a,b,c | a4=b18=1, c2=b9, ab=ba, cac-1=a-1, cbc-1=b-1 >
Smallest permutation representation of C4⋊Dic9
►Regular action on 144 points
Generators in S144
(1 106 35 127)(2 107 36 128)(3 108 19 129)(4 91 20 130)(5 92 21 131)(6 93 22 132)(7 94 23 133)(8 95 24 134)(9 96 25 135)(10 97 26 136)(11 98 27 137)(12 99 28 138)(13 100 29 139)(14 101 30 140)(15 102 31 141)(16 103 32 142)(17 104 33 143)(18 105 34 144)(37 67 117 82)(38 68 118 83)(39 69 119 84)(40 70 120 85)(41 71 121 86)(42 72 122 87)(43 55 123 88)(44 56 124 89)(45 57 125 90)(46 58 126 73)(47 59 109 74)(48 60 110 75)(49 61 111 76)(50 62 112 77)(51 63 113 78)(52 64 114 79)(53 65 115 80)(54 66 116 81)
(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 82 10 73)(2 81 11 90)(3 80 12 89)(4 79 13 88)(5 78 14 87)(6 77 15 86)(7 76 16 85)(8 75 17 84)(9 74 18 83)(19 65 28 56)(20 64 29 55)(21 63 30 72)(22 62 31 71)(23 61 32 70)(24 60 33 69)(25 59 34 68)(26 58 35 67)(27 57 36 66)(37 136 46 127)(38 135 47 144)(39 134 48 143)(40 133 49 142)(41 132 50 141)(42 131 51 140)(43 130 52 139)(44 129 53 138)(45 128 54 137)(91 114 100 123)(92 113 101 122)(93 112 102 121)(94 111 103 120)(95 110 104 119)(96 109 105 118)(97 126 106 117)(98 125 107 116)(99 124 108 115)
G:=sub<Sym(144)| (1,106,35,127)(2,107,36,128)(3,108,19,129)(4,91,20,130)(5,92,21,131)(6,93,22,132)(7,94,23,133)(8,95,24,134)(9,96,25,135)(10,97,26,136)(11,98,27,137)(12,99,28,138)(13,100,29,139)(14,101,30,140)(15,102,31,141)(16,103,32,142)(17,104,33,143)(18,105,34,144)(37,67,117,82)(38,68,118,83)(39,69,119,84)(40,70,120,85)(41,71,121,86)(42,72,122,87)(43,55,123,88)(44,56,124,89)(45,57,125,90)(46,58,126,73)(47,59,109,74)(48,60,110,75)(49,61,111,76)(50,62,112,77)(51,63,113,78)(52,64,114,79)(53,65,115,80)(54,66,116,81), (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), (1,82,10,73)(2,81,11,90)(3,80,12,89)(4,79,13,88)(5,78,14,87)(6,77,15,86)(7,76,16,85)(8,75,17,84)(9,74,18,83)(19,65,28,56)(20,64,29,55)(21,63,30,72)(22,62,31,71)(23,61,32,70)(24,60,33,69)(25,59,34,68)(26,58,35,67)(27,57,36,66)(37,136,46,127)(38,135,47,144)(39,134,48,143)(40,133,49,142)(41,132,50,141)(42,131,51,140)(43,130,52,139)(44,129,53,138)(45,128,54,137)(91,114,100,123)(92,113,101,122)(93,112,102,121)(94,111,103,120)(95,110,104,119)(96,109,105,118)(97,126,106,117)(98,125,107,116)(99,124,108,115)>;
G:=Group( (1,106,35,127)(2,107,36,128)(3,108,19,129)(4,91,20,130)(5,92,21,131)(6,93,22,132)(7,94,23,133)(8,95,24,134)(9,96,25,135)(10,97,26,136)(11,98,27,137)(12,99,28,138)(13,100,29,139)(14,101,30,140)(15,102,31,141)(16,103,32,142)(17,104,33,143)(18,105,34,144)(37,67,117,82)(38,68,118,83)(39,69,119,84)(40,70,120,85)(41,71,121,86)(42,72,122,87)(43,55,123,88)(44,56,124,89)(45,57,125,90)(46,58,126,73)(47,59,109,74)(48,60,110,75)(49,61,111,76)(50,62,112,77)(51,63,113,78)(52,64,114,79)(53,65,115,80)(54,66,116,81), (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), (1,82,10,73)(2,81,11,90)(3,80,12,89)(4,79,13,88)(5,78,14,87)(6,77,15,86)(7,76,16,85)(8,75,17,84)(9,74,18,83)(19,65,28,56)(20,64,29,55)(21,63,30,72)(22,62,31,71)(23,61,32,70)(24,60,33,69)(25,59,34,68)(26,58,35,67)(27,57,36,66)(37,136,46,127)(38,135,47,144)(39,134,48,143)(40,133,49,142)(41,132,50,141)(42,131,51,140)(43,130,52,139)(44,129,53,138)(45,128,54,137)(91,114,100,123)(92,113,101,122)(93,112,102,121)(94,111,103,120)(95,110,104,119)(96,109,105,118)(97,126,106,117)(98,125,107,116)(99,124,108,115) );
G=PermutationGroup([[(1,106,35,127),(2,107,36,128),(3,108,19,129),(4,91,20,130),(5,92,21,131),(6,93,22,132),(7,94,23,133),(8,95,24,134),(9,96,25,135),(10,97,26,136),(11,98,27,137),(12,99,28,138),(13,100,29,139),(14,101,30,140),(15,102,31,141),(16,103,32,142),(17,104,33,143),(18,105,34,144),(37,67,117,82),(38,68,118,83),(39,69,119,84),(40,70,120,85),(41,71,121,86),(42,72,122,87),(43,55,123,88),(44,56,124,89),(45,57,125,90),(46,58,126,73),(47,59,109,74),(48,60,110,75),(49,61,111,76),(50,62,112,77),(51,63,113,78),(52,64,114,79),(53,65,115,80),(54,66,116,81)], [(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)], [(1,82,10,73),(2,81,11,90),(3,80,12,89),(4,79,13,88),(5,78,14,87),(6,77,15,86),(7,76,16,85),(8,75,17,84),(9,74,18,83),(19,65,28,56),(20,64,29,55),(21,63,30,72),(22,62,31,71),(23,61,32,70),(24,60,33,69),(25,59,34,68),(26,58,35,67),(27,57,36,66),(37,136,46,127),(38,135,47,144),(39,134,48,143),(40,133,49,142),(41,132,50,141),(42,131,51,140),(43,130,52,139),(44,129,53,138),(45,128,54,137),(91,114,100,123),(92,113,101,122),(93,112,102,121),(94,111,103,120),(95,110,104,119),(96,109,105,118),(97,126,106,117),(98,125,107,116),(99,124,108,115)]])
C4⋊Dic9 is a maximal subgroup of
C36.Q8 C4.Dic18 C36.45D4 C8⋊Dic9 C72⋊1C4 C2.D72 D4⋊Dic9 Q8⋊2Dic9 C4×Dic18 C36⋊2Q8 C36.6Q8 C4×D36 C22⋊2Dic18 C23.8D18 C23.9D18 C22.4D36 C36⋊Q8 Dic9.Q8 C36.3Q8 C4⋊C4×D9 C4⋊C4⋊7D9 D18⋊2Q8 C4⋊C4⋊D9 C36.49D4 C23.26D18 C36⋊7D4 D4×Dic9 C36⋊2D4 Q8×Dic9 D18⋊3Q8 C4⋊Dic27 Dic3⋊Dic9 C36⋊C12 C36⋊Dic3
C4⋊Dic9 is a maximal quotient of
C36⋊C8 C72.C4 C8⋊Dic9 C72⋊1C4 C18.C42 C4⋊Dic27 Dic3⋊Dic9 C36⋊Dic3
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | D9 | Dic6 | D12 | Dic9 | D18 | Dic18 | D36 |
| kernel | C4⋊Dic9 | C2×Dic9 | C2×C36 | C36 | C2×C12 | C18 | C18 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 |
Matrix representation of C4⋊Dic9 ►in GL3(𝔽37) generated by
| 1 | 0 | 0 |
| 0 | 31 | 0 |
| 0 | 0 | 6 |
| 36 | 0 | 0 |
| 0 | 33 | 0 |
| 0 | 0 | 9 |
| 31 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(37))| [1,0,0,0,31,0,0,0,6],[36,0,0,0,33,0,0,0,9],[31,0,0,0,0,1,0,1,0] >;
C4⋊Dic9 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_9 % in TeX
G:=Group("C4:Dic9"); // GroupNames label
G:=SmallGroup(144,13);
// by ID
G=gap.SmallGroup(144,13);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,55,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^4=b^18=1,c^2=b^9,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
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