metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.49D4, C22⋊3Dic18, C23.25D18, (C2×C18)⋊3Q8, C4⋊Dic9⋊8C2, C9⋊4(C22⋊Q8), C18.8(C2×Q8), Dic9⋊C4⋊2C2, (C2×C4).68D18, C18.40(C2×D4), (C22×C4).7D9, (C2×Dic18)⋊6C2, (C2×C12).342D6, C4.23(C9⋊D4), (C22×C36).6C2, (C2×C6).15Dic6, C2.9(C2×Dic18), C6.35(C2×Dic6), C6.85(C4○D12), C18.15(C4○D4), (C2×C36).75C22, (C2×C18).42C23, (C22×C12).18S3, C3.(C12.48D4), (C22×C6).136D6, C12.110(C3⋊D4), C18.D4.4C2, C2.17(D36⋊5C2), C22.54(C22×D9), (C22×C18).34C22, (C2×Dic9).12C22, C2.5(C2×C9⋊D4), C6.87(C2×C3⋊D4), (C2×C6).199(C22×S3), SmallGroup(288,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.49D4
G = < a,b,c | a36=b4=1, c2=a18, bab-1=cac-1=a-1, cbc-1=a18b-1 >
Subgroups: 380 in 111 conjugacy classes, 50 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C9, Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C18 [×3], C18 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C22⋊Q8, Dic9 [×4], C36 [×2], C36, C2×C18, C2×C18 [×2], C2×C18 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, Dic18 [×2], C2×Dic9 [×4], C2×C36 [×2], C2×C36 [×2], C22×C18, C12.48D4, Dic9⋊C4 [×2], C4⋊Dic9, C18.D4 [×2], C2×Dic18, C22×C36, C36.49D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, D9, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, D18 [×3], C2×Dic6, C4○D12, C2×C3⋊D4, Dic18 [×2], C9⋊D4 [×2], C22×D9, C12.48D4, C2×Dic18, D36⋊5C2, C2×C9⋊D4, C36.49D4
►On 144 points
(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 45 100 114)(2 44 101 113)(3 43 102 112)(4 42 103 111)(5 41 104 110)(6 40 105 109)(7 39 106 144)(8 38 107 143)(9 37 108 142)(10 72 73 141)(11 71 74 140)(12 70 75 139)(13 69 76 138)(14 68 77 137)(15 67 78 136)(16 66 79 135)(17 65 80 134)(18 64 81 133)(19 63 82 132)(20 62 83 131)(21 61 84 130)(22 60 85 129)(23 59 86 128)(24 58 87 127)(25 57 88 126)(26 56 89 125)(27 55 90 124)(28 54 91 123)(29 53 92 122)(30 52 93 121)(31 51 94 120)(32 50 95 119)(33 49 96 118)(34 48 97 117)(35 47 98 116)(36 46 99 115)
(1 132 19 114)(2 131 20 113)(3 130 21 112)(4 129 22 111)(5 128 23 110)(6 127 24 109)(7 126 25 144)(8 125 26 143)(9 124 27 142)(10 123 28 141)(11 122 29 140)(12 121 30 139)(13 120 31 138)(14 119 32 137)(15 118 33 136)(16 117 34 135)(17 116 35 134)(18 115 36 133)(37 108 55 90)(38 107 56 89)(39 106 57 88)(40 105 58 87)(41 104 59 86)(42 103 60 85)(43 102 61 84)(44 101 62 83)(45 100 63 82)(46 99 64 81)(47 98 65 80)(48 97 66 79)(49 96 67 78)(50 95 68 77)(51 94 69 76)(52 93 70 75)(53 92 71 74)(54 91 72 73)
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), (1,45,100,114)(2,44,101,113)(3,43,102,112)(4,42,103,111)(5,41,104,110)(6,40,105,109)(7,39,106,144)(8,38,107,143)(9,37,108,142)(10,72,73,141)(11,71,74,140)(12,70,75,139)(13,69,76,138)(14,68,77,137)(15,67,78,136)(16,66,79,135)(17,65,80,134)(18,64,81,133)(19,63,82,132)(20,62,83,131)(21,61,84,130)(22,60,85,129)(23,59,86,128)(24,58,87,127)(25,57,88,126)(26,56,89,125)(27,55,90,124)(28,54,91,123)(29,53,92,122)(30,52,93,121)(31,51,94,120)(32,50,95,119)(33,49,96,118)(34,48,97,117)(35,47,98,116)(36,46,99,115), (1,132,19,114)(2,131,20,113)(3,130,21,112)(4,129,22,111)(5,128,23,110)(6,127,24,109)(7,126,25,144)(8,125,26,143)(9,124,27,142)(10,123,28,141)(11,122,29,140)(12,121,30,139)(13,120,31,138)(14,119,32,137)(15,118,33,136)(16,117,34,135)(17,116,35,134)(18,115,36,133)(37,108,55,90)(38,107,56,89)(39,106,57,88)(40,105,58,87)(41,104,59,86)(42,103,60,85)(43,102,61,84)(44,101,62,83)(45,100,63,82)(46,99,64,81)(47,98,65,80)(48,97,66,79)(49,96,67,78)(50,95,68,77)(51,94,69,76)(52,93,70,75)(53,92,71,74)(54,91,72,73)>;
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), (1,45,100,114)(2,44,101,113)(3,43,102,112)(4,42,103,111)(5,41,104,110)(6,40,105,109)(7,39,106,144)(8,38,107,143)(9,37,108,142)(10,72,73,141)(11,71,74,140)(12,70,75,139)(13,69,76,138)(14,68,77,137)(15,67,78,136)(16,66,79,135)(17,65,80,134)(18,64,81,133)(19,63,82,132)(20,62,83,131)(21,61,84,130)(22,60,85,129)(23,59,86,128)(24,58,87,127)(25,57,88,126)(26,56,89,125)(27,55,90,124)(28,54,91,123)(29,53,92,122)(30,52,93,121)(31,51,94,120)(32,50,95,119)(33,49,96,118)(34,48,97,117)(35,47,98,116)(36,46,99,115), (1,132,19,114)(2,131,20,113)(3,130,21,112)(4,129,22,111)(5,128,23,110)(6,127,24,109)(7,126,25,144)(8,125,26,143)(9,124,27,142)(10,123,28,141)(11,122,29,140)(12,121,30,139)(13,120,31,138)(14,119,32,137)(15,118,33,136)(16,117,34,135)(17,116,35,134)(18,115,36,133)(37,108,55,90)(38,107,56,89)(39,106,57,88)(40,105,58,87)(41,104,59,86)(42,103,60,85)(43,102,61,84)(44,101,62,83)(45,100,63,82)(46,99,64,81)(47,98,65,80)(48,97,66,79)(49,96,67,78)(50,95,68,77)(51,94,69,76)(52,93,70,75)(53,92,71,74)(54,91,72,73) );
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)], [(1,45,100,114),(2,44,101,113),(3,43,102,112),(4,42,103,111),(5,41,104,110),(6,40,105,109),(7,39,106,144),(8,38,107,143),(9,37,108,142),(10,72,73,141),(11,71,74,140),(12,70,75,139),(13,69,76,138),(14,68,77,137),(15,67,78,136),(16,66,79,135),(17,65,80,134),(18,64,81,133),(19,63,82,132),(20,62,83,131),(21,61,84,130),(22,60,85,129),(23,59,86,128),(24,58,87,127),(25,57,88,126),(26,56,89,125),(27,55,90,124),(28,54,91,123),(29,53,92,122),(30,52,93,121),(31,51,94,120),(32,50,95,119),(33,49,96,118),(34,48,97,117),(35,47,98,116),(36,46,99,115)], [(1,132,19,114),(2,131,20,113),(3,130,21,112),(4,129,22,111),(5,128,23,110),(6,127,24,109),(7,126,25,144),(8,125,26,143),(9,124,27,142),(10,123,28,141),(11,122,29,140),(12,121,30,139),(13,120,31,138),(14,119,32,137),(15,118,33,136),(16,117,34,135),(17,116,35,134),(18,115,36,133),(37,108,55,90),(38,107,56,89),(39,106,57,88),(40,105,58,87),(41,104,59,86),(42,103,60,85),(43,102,61,84),(44,101,62,83),(45,100,63,82),(46,99,64,81),(47,98,65,80),(48,97,66,79),(49,96,67,78),(50,95,68,77),(51,94,69,76),(52,93,70,75),(53,92,71,74),(54,91,72,73)])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | Dic6 | D18 | D18 | C4○D12 | C9⋊D4 | Dic18 | D36⋊5C2 |
| kernel | C36.49D4 | Dic9⋊C4 | C4⋊Dic9 | C18.D4 | C2×Dic18 | C22×C36 | C22×C12 | C36 | C2×C18 | C2×C12 | C22×C6 | C18 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 3 | 4 | 4 | 6 | 3 | 4 | 12 | 12 | 12 |
Matrix representation of C36.49D4 ►in GL4(𝔽37) generated by
| 8 | 35 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 20 |
| 36 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 1 | 13 | 0 | 0 |
| 34 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [8,0,0,0,35,14,0,0,0,0,13,0,0,0,0,20],[36,3,0,0,0,1,0,0,0,0,0,36,0,0,1,0],[1,34,0,0,13,36,0,0,0,0,0,36,0,0,1,0] >;
C36.49D4 in GAP, Magma, Sage, TeX
C_{36}._{49}D_4 % in TeX
G:=Group("C36.49D4"); // GroupNames label
G:=SmallGroup(288,134);
// by ID
G=gap.SmallGroup(288,134);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,254,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=1,c^2=a^18,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^18*b^-1>;
// generators/relations