metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.2D6, C8.1D18, C12.8D12, C36.13D4, C4.15D36, Dic36⋊2C2, M4(2)⋊2D9, C72.1C22, C22.6D36, C36.33C23, D36.8C22, Dic18.8C22, C72⋊C2⋊2C2, (C2×C6).7D12, (C2×C18).6D4, (C2×C12).54D6, C2.16(C2×D36), (C2×C4).13D18, C18.14(C2×D4), C6.43(C2×D12), C9⋊1(C8.C22), C3.(C8.D6), (C2×Dic18)⋊8C2, (C9×M4(2))⋊2C2, C4.31(C22×D9), D36⋊5C2.4C2, (C2×C36).32C22, (C3×M4(2)).2S3, C12.184(C22×S3), SmallGroup(288,119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D18
G = < a,b,c | a8=1, b18=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b17 >
Subgroups: 436 in 90 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, D9, C18, C18, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, Dic9, C36, D18, C2×C18, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, C72, Dic18, Dic18, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, C8.D6, Dic36, C72⋊C2, C9×M4(2), C2×Dic18, D36⋊5C2, C8.D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, C8.C22, D18, C2×D12, D36, C22×D9, C8.D6, C2×D36, C8.D18
►On 144 points
Generators in S144
(1 57 84 127 19 39 102 109)(2 40 85 110 20 58 103 128)(3 59 86 129 21 41 104 111)(4 42 87 112 22 60 105 130)(5 61 88 131 23 43 106 113)(6 44 89 114 24 62 107 132)(7 63 90 133 25 45 108 115)(8 46 91 116 26 64 73 134)(9 65 92 135 27 47 74 117)(10 48 93 118 28 66 75 136)(11 67 94 137 29 49 76 119)(12 50 95 120 30 68 77 138)(13 69 96 139 31 51 78 121)(14 52 97 122 32 70 79 140)(15 71 98 141 33 53 80 123)(16 54 99 124 34 72 81 142)(17 37 100 143 35 55 82 125)(18 56 101 126 36 38 83 144)
(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 18 19 36)(2 35 20 17)(3 16 21 34)(4 33 22 15)(5 14 23 32)(6 31 24 13)(7 12 25 30)(8 29 26 11)(9 10 27 28)(37 128 55 110)(38 109 56 127)(39 126 57 144)(40 143 58 125)(41 124 59 142)(42 141 60 123)(43 122 61 140)(44 139 62 121)(45 120 63 138)(46 137 64 119)(47 118 65 136)(48 135 66 117)(49 116 67 134)(50 133 68 115)(51 114 69 132)(52 131 70 113)(53 112 71 130)(54 129 72 111)(73 76 91 94)(74 93 92 75)(77 108 95 90)(78 89 96 107)(79 106 97 88)(80 87 98 105)(81 104 99 86)(82 85 100 103)(83 102 101 84)
G:=sub<Sym(144)| (1,57,84,127,19,39,102,109)(2,40,85,110,20,58,103,128)(3,59,86,129,21,41,104,111)(4,42,87,112,22,60,105,130)(5,61,88,131,23,43,106,113)(6,44,89,114,24,62,107,132)(7,63,90,133,25,45,108,115)(8,46,91,116,26,64,73,134)(9,65,92,135,27,47,74,117)(10,48,93,118,28,66,75,136)(11,67,94,137,29,49,76,119)(12,50,95,120,30,68,77,138)(13,69,96,139,31,51,78,121)(14,52,97,122,32,70,79,140)(15,71,98,141,33,53,80,123)(16,54,99,124,34,72,81,142)(17,37,100,143,35,55,82,125)(18,56,101,126,36,38,83,144), (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), (1,18,19,36)(2,35,20,17)(3,16,21,34)(4,33,22,15)(5,14,23,32)(6,31,24,13)(7,12,25,30)(8,29,26,11)(9,10,27,28)(37,128,55,110)(38,109,56,127)(39,126,57,144)(40,143,58,125)(41,124,59,142)(42,141,60,123)(43,122,61,140)(44,139,62,121)(45,120,63,138)(46,137,64,119)(47,118,65,136)(48,135,66,117)(49,116,67,134)(50,133,68,115)(51,114,69,132)(52,131,70,113)(53,112,71,130)(54,129,72,111)(73,76,91,94)(74,93,92,75)(77,108,95,90)(78,89,96,107)(79,106,97,88)(80,87,98,105)(81,104,99,86)(82,85,100,103)(83,102,101,84)>;
G:=Group( (1,57,84,127,19,39,102,109)(2,40,85,110,20,58,103,128)(3,59,86,129,21,41,104,111)(4,42,87,112,22,60,105,130)(5,61,88,131,23,43,106,113)(6,44,89,114,24,62,107,132)(7,63,90,133,25,45,108,115)(8,46,91,116,26,64,73,134)(9,65,92,135,27,47,74,117)(10,48,93,118,28,66,75,136)(11,67,94,137,29,49,76,119)(12,50,95,120,30,68,77,138)(13,69,96,139,31,51,78,121)(14,52,97,122,32,70,79,140)(15,71,98,141,33,53,80,123)(16,54,99,124,34,72,81,142)(17,37,100,143,35,55,82,125)(18,56,101,126,36,38,83,144), (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), (1,18,19,36)(2,35,20,17)(3,16,21,34)(4,33,22,15)(5,14,23,32)(6,31,24,13)(7,12,25,30)(8,29,26,11)(9,10,27,28)(37,128,55,110)(38,109,56,127)(39,126,57,144)(40,143,58,125)(41,124,59,142)(42,141,60,123)(43,122,61,140)(44,139,62,121)(45,120,63,138)(46,137,64,119)(47,118,65,136)(48,135,66,117)(49,116,67,134)(50,133,68,115)(51,114,69,132)(52,131,70,113)(53,112,71,130)(54,129,72,111)(73,76,91,94)(74,93,92,75)(77,108,95,90)(78,89,96,107)(79,106,97,88)(80,87,98,105)(81,104,99,86)(82,85,100,103)(83,102,101,84) );
G=PermutationGroup([[(1,57,84,127,19,39,102,109),(2,40,85,110,20,58,103,128),(3,59,86,129,21,41,104,111),(4,42,87,112,22,60,105,130),(5,61,88,131,23,43,106,113),(6,44,89,114,24,62,107,132),(7,63,90,133,25,45,108,115),(8,46,91,116,26,64,73,134),(9,65,92,135,27,47,74,117),(10,48,93,118,28,66,75,136),(11,67,94,137,29,49,76,119),(12,50,95,120,30,68,77,138),(13,69,96,139,31,51,78,121),(14,52,97,122,32,70,79,140),(15,71,98,141,33,53,80,123),(16,54,99,124,34,72,81,142),(17,37,100,143,35,55,82,125),(18,56,101,126,36,38,83,144)], [(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)], [(1,18,19,36),(2,35,20,17),(3,16,21,34),(4,33,22,15),(5,14,23,32),(6,31,24,13),(7,12,25,30),(8,29,26,11),(9,10,27,28),(37,128,55,110),(38,109,56,127),(39,126,57,144),(40,143,58,125),(41,124,59,142),(42,141,60,123),(43,122,61,140),(44,139,62,121),(45,120,63,138),(46,137,64,119),(47,118,65,136),(48,135,66,117),(49,116,67,134),(50,133,68,115),(51,114,69,132),(52,131,70,113),(53,112,71,130),(54,129,72,111),(73,76,91,94),(74,93,92,75),(77,108,95,90),(78,89,96,107),(79,106,97,88),(80,87,98,105),(81,104,99,86),(82,85,100,103),(83,102,101,84)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | 36H | 36I | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 36 | 2 | 2 | 2 | 36 | 36 | 36 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | D12 | D12 | D18 | D18 | D36 | D36 | C8.C22 | C8.D6 | C8.D18 |
| kernel | C8.D18 | Dic36 | C72⋊C2 | C9×M4(2) | C2×Dic18 | D36⋊5C2 | C3×M4(2) | C36 | C2×C18 | C24 | C2×C12 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of C8.D18 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 5 | 47 | 0 |
| 0 | 0 | 34 | 5 | 60 | 13 |
| 0 | 0 | 13 | 21 | 47 | 39 |
| 0 | 0 | 34 | 34 | 47 | 34 |
| 70 | 45 | 0 | 0 | 0 | 0 |
| 28 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 0 | 68 | 68 |
| 0 | 0 | 34 | 60 | 68 | 0 |
| 0 | 0 | 52 | 34 | 52 | 39 |
| 0 | 0 | 13 | 39 | 39 | 52 |
| 3 | 42 | 0 | 0 | 0 | 0 |
| 45 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 0 | 68 | 68 |
| 0 | 0 | 21 | 13 | 0 | 68 |
| 0 | 0 | 13 | 39 | 39 | 52 |
| 0 | 0 | 52 | 34 | 52 | 39 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,60,34,13,34,0,0,5,5,21,34,0,0,47,60,47,47,0,0,0,13,39,34],[70,28,0,0,0,0,45,42,0,0,0,0,0,0,55,34,52,13,0,0,0,60,34,39,0,0,68,68,52,39,0,0,68,0,39,52],[3,45,0,0,0,0,42,70,0,0,0,0,0,0,55,21,13,52,0,0,0,13,39,34,0,0,68,0,39,52,0,0,68,68,52,39] >;
C8.D18 in GAP, Magma, Sage, TeX
C_8.D_{18} % in TeX
G:=Group("C8.D18"); // GroupNames label
G:=SmallGroup(288,119);
// by ID
G=gap.SmallGroup(288,119);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^18=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^17>;
// generators/relations