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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C9, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊Q8, Dic9, C36, C36, C2×C18, C2×C18, C2×C18, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, Dic18, C2×Dic9, C2×C36, C2×C36, C22×C18, C12.48D4, Dic9⋊C4, C4⋊Dic9, C18.D4, C2×Dic18, C22×C36, C36.49D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D9, Dic6, C3⋊D4, C22×S3, C22⋊Q8, D18, C2×Dic6, C4○D12, C2×C3⋊D4, Dic18, C9⋊D4, C22×D9, C12.48D4, C2×Dic18, D36⋊5C2, C2×C9⋊D4, C36.49D4
►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 50 86 144)(2 49 87 143)(3 48 88 142)(4 47 89 141)(5 46 90 140)(6 45 91 139)(7 44 92 138)(8 43 93 137)(9 42 94 136)(10 41 95 135)(11 40 96 134)(12 39 97 133)(13 38 98 132)(14 37 99 131)(15 72 100 130)(16 71 101 129)(17 70 102 128)(18 69 103 127)(19 68 104 126)(20 67 105 125)(21 66 106 124)(22 65 107 123)(23 64 108 122)(24 63 73 121)(25 62 74 120)(26 61 75 119)(27 60 76 118)(28 59 77 117)(29 58 78 116)(30 57 79 115)(31 56 80 114)(32 55 81 113)(33 54 82 112)(34 53 83 111)(35 52 84 110)(36 51 85 109)
(1 126 19 144)(2 125 20 143)(3 124 21 142)(4 123 22 141)(5 122 23 140)(6 121 24 139)(7 120 25 138)(8 119 26 137)(9 118 27 136)(10 117 28 135)(11 116 29 134)(12 115 30 133)(13 114 31 132)(14 113 32 131)(15 112 33 130)(16 111 34 129)(17 110 35 128)(18 109 36 127)(37 99 55 81)(38 98 56 80)(39 97 57 79)(40 96 58 78)(41 95 59 77)(42 94 60 76)(43 93 61 75)(44 92 62 74)(45 91 63 73)(46 90 64 108)(47 89 65 107)(48 88 66 106)(49 87 67 105)(50 86 68 104)(51 85 69 103)(52 84 70 102)(53 83 71 101)(54 82 72 100)
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), (1,50,86,144)(2,49,87,143)(3,48,88,142)(4,47,89,141)(5,46,90,140)(6,45,91,139)(7,44,92,138)(8,43,93,137)(9,42,94,136)(10,41,95,135)(11,40,96,134)(12,39,97,133)(13,38,98,132)(14,37,99,131)(15,72,100,130)(16,71,101,129)(17,70,102,128)(18,69,103,127)(19,68,104,126)(20,67,105,125)(21,66,106,124)(22,65,107,123)(23,64,108,122)(24,63,73,121)(25,62,74,120)(26,61,75,119)(27,60,76,118)(28,59,77,117)(29,58,78,116)(30,57,79,115)(31,56,80,114)(32,55,81,113)(33,54,82,112)(34,53,83,111)(35,52,84,110)(36,51,85,109), (1,126,19,144)(2,125,20,143)(3,124,21,142)(4,123,22,141)(5,122,23,140)(6,121,24,139)(7,120,25,138)(8,119,26,137)(9,118,27,136)(10,117,28,135)(11,116,29,134)(12,115,30,133)(13,114,31,132)(14,113,32,131)(15,112,33,130)(16,111,34,129)(17,110,35,128)(18,109,36,127)(37,99,55,81)(38,98,56,80)(39,97,57,79)(40,96,58,78)(41,95,59,77)(42,94,60,76)(43,93,61,75)(44,92,62,74)(45,91,63,73)(46,90,64,108)(47,89,65,107)(48,88,66,106)(49,87,67,105)(50,86,68,104)(51,85,69,103)(52,84,70,102)(53,83,71,101)(54,82,72,100)>;
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), (1,50,86,144)(2,49,87,143)(3,48,88,142)(4,47,89,141)(5,46,90,140)(6,45,91,139)(7,44,92,138)(8,43,93,137)(9,42,94,136)(10,41,95,135)(11,40,96,134)(12,39,97,133)(13,38,98,132)(14,37,99,131)(15,72,100,130)(16,71,101,129)(17,70,102,128)(18,69,103,127)(19,68,104,126)(20,67,105,125)(21,66,106,124)(22,65,107,123)(23,64,108,122)(24,63,73,121)(25,62,74,120)(26,61,75,119)(27,60,76,118)(28,59,77,117)(29,58,78,116)(30,57,79,115)(31,56,80,114)(32,55,81,113)(33,54,82,112)(34,53,83,111)(35,52,84,110)(36,51,85,109), (1,126,19,144)(2,125,20,143)(3,124,21,142)(4,123,22,141)(5,122,23,140)(6,121,24,139)(7,120,25,138)(8,119,26,137)(9,118,27,136)(10,117,28,135)(11,116,29,134)(12,115,30,133)(13,114,31,132)(14,113,32,131)(15,112,33,130)(16,111,34,129)(17,110,35,128)(18,109,36,127)(37,99,55,81)(38,98,56,80)(39,97,57,79)(40,96,58,78)(41,95,59,77)(42,94,60,76)(43,93,61,75)(44,92,62,74)(45,91,63,73)(46,90,64,108)(47,89,65,107)(48,88,66,106)(49,87,67,105)(50,86,68,104)(51,85,69,103)(52,84,70,102)(53,83,71,101)(54,82,72,100) );
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)], [(1,50,86,144),(2,49,87,143),(3,48,88,142),(4,47,89,141),(5,46,90,140),(6,45,91,139),(7,44,92,138),(8,43,93,137),(9,42,94,136),(10,41,95,135),(11,40,96,134),(12,39,97,133),(13,38,98,132),(14,37,99,131),(15,72,100,130),(16,71,101,129),(17,70,102,128),(18,69,103,127),(19,68,104,126),(20,67,105,125),(21,66,106,124),(22,65,107,123),(23,64,108,122),(24,63,73,121),(25,62,74,120),(26,61,75,119),(27,60,76,118),(28,59,77,117),(29,58,78,116),(30,57,79,115),(31,56,80,114),(32,55,81,113),(33,54,82,112),(34,53,83,111),(35,52,84,110),(36,51,85,109)], [(1,126,19,144),(2,125,20,143),(3,124,21,142),(4,123,22,141),(5,122,23,140),(6,121,24,139),(7,120,25,138),(8,119,26,137),(9,118,27,136),(10,117,28,135),(11,116,29,134),(12,115,30,133),(13,114,31,132),(14,113,32,131),(15,112,33,130),(16,111,34,129),(17,110,35,128),(18,109,36,127),(37,99,55,81),(38,98,56,80),(39,97,57,79),(40,96,58,78),(41,95,59,77),(42,94,60,76),(43,93,61,75),(44,92,62,74),(45,91,63,73),(46,90,64,108),(47,89,65,107),(48,88,66,106),(49,87,67,105),(50,86,68,104),(51,85,69,103),(52,84,70,102),(53,83,71,101),(54,82,72,100)]])
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