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 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C9, Dic3 [×3], C12 [×2], D6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, D9, C18, C18, C24 [×2], Dic6 [×4], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C8.C22, Dic9 [×3], C36 [×2], D18, C2×C18, C24⋊C2 [×2], Dic12 [×2], C3×M4(2), C2×Dic6, C4○D12, C72 [×2], Dic18, Dic18 [×2], Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, C8.D6, Dic36 [×2], C72⋊C2 [×2], C9×M4(2), C2×Dic18, D36⋊5C2, C8.D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, D12 [×2], C22×S3, C8.C22, D18 [×3], C2×D12, D36 [×2], C22×D9, C8.D6, C2×D36, C8.D18
►On 144 points
Generators in S144
(1 62 103 126 19 44 85 144)(2 45 104 109 20 63 86 127)(3 64 105 128 21 46 87 110)(4 47 106 111 22 65 88 129)(5 66 107 130 23 48 89 112)(6 49 108 113 24 67 90 131)(7 68 73 132 25 50 91 114)(8 51 74 115 26 69 92 133)(9 70 75 134 27 52 93 116)(10 53 76 117 28 71 94 135)(11 72 77 136 29 54 95 118)(12 55 78 119 30 37 96 137)(13 38 79 138 31 56 97 120)(14 57 80 121 32 39 98 139)(15 40 81 140 33 58 99 122)(16 59 82 123 34 41 100 141)(17 42 83 142 35 60 101 124)(18 61 84 125 36 43 102 143)
(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)(37 114 55 132)(38 131 56 113)(39 112 57 130)(40 129 58 111)(41 110 59 128)(42 127 60 109)(43 144 61 126)(44 125 62 143)(45 142 63 124)(46 123 64 141)(47 140 65 122)(48 121 66 139)(49 138 67 120)(50 119 68 137)(51 136 69 118)(52 117 70 135)(53 134 71 116)(54 115 72 133)(73 96 91 78)(74 77 92 95)(75 94 93 76)(79 90 97 108)(80 107 98 89)(81 88 99 106)(82 105 100 87)(83 86 101 104)(84 103 102 85)
G:=sub<Sym(144)| (1,62,103,126,19,44,85,144)(2,45,104,109,20,63,86,127)(3,64,105,128,21,46,87,110)(4,47,106,111,22,65,88,129)(5,66,107,130,23,48,89,112)(6,49,108,113,24,67,90,131)(7,68,73,132,25,50,91,114)(8,51,74,115,26,69,92,133)(9,70,75,134,27,52,93,116)(10,53,76,117,28,71,94,135)(11,72,77,136,29,54,95,118)(12,55,78,119,30,37,96,137)(13,38,79,138,31,56,97,120)(14,57,80,121,32,39,98,139)(15,40,81,140,33,58,99,122)(16,59,82,123,34,41,100,141)(17,42,83,142,35,60,101,124)(18,61,84,125,36,43,102,143), (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)(37,114,55,132)(38,131,56,113)(39,112,57,130)(40,129,58,111)(41,110,59,128)(42,127,60,109)(43,144,61,126)(44,125,62,143)(45,142,63,124)(46,123,64,141)(47,140,65,122)(48,121,66,139)(49,138,67,120)(50,119,68,137)(51,136,69,118)(52,117,70,135)(53,134,71,116)(54,115,72,133)(73,96,91,78)(74,77,92,95)(75,94,93,76)(79,90,97,108)(80,107,98,89)(81,88,99,106)(82,105,100,87)(83,86,101,104)(84,103,102,85)>;
G:=Group( (1,62,103,126,19,44,85,144)(2,45,104,109,20,63,86,127)(3,64,105,128,21,46,87,110)(4,47,106,111,22,65,88,129)(5,66,107,130,23,48,89,112)(6,49,108,113,24,67,90,131)(7,68,73,132,25,50,91,114)(8,51,74,115,26,69,92,133)(9,70,75,134,27,52,93,116)(10,53,76,117,28,71,94,135)(11,72,77,136,29,54,95,118)(12,55,78,119,30,37,96,137)(13,38,79,138,31,56,97,120)(14,57,80,121,32,39,98,139)(15,40,81,140,33,58,99,122)(16,59,82,123,34,41,100,141)(17,42,83,142,35,60,101,124)(18,61,84,125,36,43,102,143), (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)(37,114,55,132)(38,131,56,113)(39,112,57,130)(40,129,58,111)(41,110,59,128)(42,127,60,109)(43,144,61,126)(44,125,62,143)(45,142,63,124)(46,123,64,141)(47,140,65,122)(48,121,66,139)(49,138,67,120)(50,119,68,137)(51,136,69,118)(52,117,70,135)(53,134,71,116)(54,115,72,133)(73,96,91,78)(74,77,92,95)(75,94,93,76)(79,90,97,108)(80,107,98,89)(81,88,99,106)(82,105,100,87)(83,86,101,104)(84,103,102,85) );
G=PermutationGroup([(1,62,103,126,19,44,85,144),(2,45,104,109,20,63,86,127),(3,64,105,128,21,46,87,110),(4,47,106,111,22,65,88,129),(5,66,107,130,23,48,89,112),(6,49,108,113,24,67,90,131),(7,68,73,132,25,50,91,114),(8,51,74,115,26,69,92,133),(9,70,75,134,27,52,93,116),(10,53,76,117,28,71,94,135),(11,72,77,136,29,54,95,118),(12,55,78,119,30,37,96,137),(13,38,79,138,31,56,97,120),(14,57,80,121,32,39,98,139),(15,40,81,140,33,58,99,122),(16,59,82,123,34,41,100,141),(17,42,83,142,35,60,101,124),(18,61,84,125,36,43,102,143)], [(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),(37,114,55,132),(38,131,56,113),(39,112,57,130),(40,129,58,111),(41,110,59,128),(42,127,60,109),(43,144,61,126),(44,125,62,143),(45,142,63,124),(46,123,64,141),(47,140,65,122),(48,121,66,139),(49,138,67,120),(50,119,68,137),(51,136,69,118),(52,117,70,135),(53,134,71,116),(54,115,72,133),(73,96,91,78),(74,77,92,95),(75,94,93,76),(79,90,97,108),(80,107,98,89),(81,88,99,106),(82,105,100,87),(83,86,101,104),(84,103,102,85)])
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