metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊C8, C36.35D4, C4.19D36, C12.54D12, C18.3M4(2), (C2×C8)⋊1D9, (C2×C72)⋊1C2, C3.(D6⋊C8), C2.5(C8×D9), C6.10(S3×C8), C9⋊1(C22⋊C8), C18.5(C2×C8), (C2×C24).3S3, (C2×C4).93D18, C6.6(C8⋊S3), C2.1(D18⋊C4), C6.13(D6⋊C4), (C2×C12).407D6, C2.3(C8⋊D9), C4.27(C9⋊D4), (C2×Dic9).4C4, (C22×D9).2C4, C22.11(C4×D9), C18.6(C22⋊C4), C12.122(C3⋊D4), (C2×C36).105C22, (C2×C9⋊C8)⋊9C2, (C2×C4×D9).6C2, (C2×C6).37(C4×S3), (C2×C18).12(C2×C4), SmallGroup(288,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18⋊C8
G = < a,b,c | a18=b2=c8=1, bab=a-1, ac=ca, cbc-1=a9b >
Subgroups: 332 in 75 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4, C2×C4 [×3], C23, C9, Dic3, C12 [×2], D6 [×4], C2×C6, C2×C8, C2×C8, C22×C4, D9 [×2], C18 [×3], C3⋊C8, C24, C4×S3 [×2], C2×Dic3, C2×C12, C22×S3, C22⋊C8, Dic9, C36 [×2], D18 [×2], D18 [×2], C2×C18, C2×C3⋊C8, C2×C24, S3×C2×C4, C9⋊C8, C72, C4×D9 [×2], C2×Dic9, C2×C36, C22×D9, D6⋊C8, C2×C9⋊C8, C2×C72, C2×C4×D9, D18⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D4 [×2], D6, C22⋊C4, C2×C8, M4(2), D9, C4×S3, D12, C3⋊D4, C22⋊C8, D18, S3×C8, C8⋊S3, D6⋊C4, C4×D9, D36, C9⋊D4, D6⋊C8, C8×D9, C8⋊D9, D18⋊C4, D18⋊C8
►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 57)(2 56)(3 55)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 60)(17 59)(18 58)(19 114)(20 113)(21 112)(22 111)(23 110)(24 109)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 94)(38 93)(39 92)(40 91)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 100)(50 99)(51 98)(52 97)(53 96)(54 95)(73 141)(74 140)(75 139)(76 138)(77 137)(78 136)(79 135)(80 134)(81 133)(82 132)(83 131)(84 130)(85 129)(86 128)(87 127)(88 144)(89 143)(90 142)
(1 96 77 119 58 45 138 24)(2 97 78 120 59 46 139 25)(3 98 79 121 60 47 140 26)(4 99 80 122 61 48 141 27)(5 100 81 123 62 49 142 28)(6 101 82 124 63 50 143 29)(7 102 83 125 64 51 144 30)(8 103 84 126 65 52 127 31)(9 104 85 109 66 53 128 32)(10 105 86 110 67 54 129 33)(11 106 87 111 68 37 130 34)(12 107 88 112 69 38 131 35)(13 108 89 113 70 39 132 36)(14 91 90 114 71 40 133 19)(15 92 73 115 72 41 134 20)(16 93 74 116 55 42 135 21)(17 94 75 117 56 43 136 22)(18 95 76 118 57 44 137 23)
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), (1,57)(2,56)(3,55)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,94)(38,93)(39,92)(40,91)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(81,133)(82,132)(83,131)(84,130)(85,129)(86,128)(87,127)(88,144)(89,143)(90,142), (1,96,77,119,58,45,138,24)(2,97,78,120,59,46,139,25)(3,98,79,121,60,47,140,26)(4,99,80,122,61,48,141,27)(5,100,81,123,62,49,142,28)(6,101,82,124,63,50,143,29)(7,102,83,125,64,51,144,30)(8,103,84,126,65,52,127,31)(9,104,85,109,66,53,128,32)(10,105,86,110,67,54,129,33)(11,106,87,111,68,37,130,34)(12,107,88,112,69,38,131,35)(13,108,89,113,70,39,132,36)(14,91,90,114,71,40,133,19)(15,92,73,115,72,41,134,20)(16,93,74,116,55,42,135,21)(17,94,75,117,56,43,136,22)(18,95,76,118,57,44,137,23)>;
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), (1,57)(2,56)(3,55)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,94)(38,93)(39,92)(40,91)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(81,133)(82,132)(83,131)(84,130)(85,129)(86,128)(87,127)(88,144)(89,143)(90,142), (1,96,77,119,58,45,138,24)(2,97,78,120,59,46,139,25)(3,98,79,121,60,47,140,26)(4,99,80,122,61,48,141,27)(5,100,81,123,62,49,142,28)(6,101,82,124,63,50,143,29)(7,102,83,125,64,51,144,30)(8,103,84,126,65,52,127,31)(9,104,85,109,66,53,128,32)(10,105,86,110,67,54,129,33)(11,106,87,111,68,37,130,34)(12,107,88,112,69,38,131,35)(13,108,89,113,70,39,132,36)(14,91,90,114,71,40,133,19)(15,92,73,115,72,41,134,20)(16,93,74,116,55,42,135,21)(17,94,75,117,56,43,136,22)(18,95,76,118,57,44,137,23) );
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)], [(1,57),(2,56),(3,55),(4,72),(5,71),(6,70),(7,69),(8,68),(9,67),(10,66),(11,65),(12,64),(13,63),(14,62),(15,61),(16,60),(17,59),(18,58),(19,114),(20,113),(21,112),(22,111),(23,110),(24,109),(25,126),(26,125),(27,124),(28,123),(29,122),(30,121),(31,120),(32,119),(33,118),(34,117),(35,116),(36,115),(37,94),(38,93),(39,92),(40,91),(41,108),(42,107),(43,106),(44,105),(45,104),(46,103),(47,102),(48,101),(49,100),(50,99),(51,98),(52,97),(53,96),(54,95),(73,141),(74,140),(75,139),(76,138),(77,137),(78,136),(79,135),(80,134),(81,133),(82,132),(83,131),(84,130),(85,129),(86,128),(87,127),(88,144),(89,143),(90,142)], [(1,96,77,119,58,45,138,24),(2,97,78,120,59,46,139,25),(3,98,79,121,60,47,140,26),(4,99,80,122,61,48,141,27),(5,100,81,123,62,49,142,28),(6,101,82,124,63,50,143,29),(7,102,83,125,64,51,144,30),(8,103,84,126,65,52,127,31),(9,104,85,109,66,53,128,32),(10,105,86,110,67,54,129,33),(11,106,87,111,68,37,130,34),(12,107,88,112,69,38,131,35),(13,108,89,113,70,39,132,36),(14,91,90,114,71,40,133,19),(15,92,73,115,72,41,134,20),(16,93,74,116,55,42,135,21),(17,94,75,117,56,43,136,22),(18,95,76,118,57,44,137,23)])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D9 | D12 | C3⋊D4 | C4×S3 | D18 | S3×C8 | C8⋊S3 | D36 | C9⋊D4 | C4×D9 | C8×D9 | C8⋊D9 |
| kernel | D18⋊C8 | C2×C9⋊C8 | C2×C72 | C2×C4×D9 | C2×Dic9 | C22×D9 | D18 | C2×C24 | C36 | C2×C12 | C18 | C2×C8 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of D18⋊C8 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 3 | 28 |
| 0 | 45 | 31 |
| 1 | 0 | 0 |
| 0 | 28 | 42 |
| 0 | 70 | 45 |
| 10 | 0 | 0 |
| 0 | 65 | 16 |
| 0 | 57 | 8 |
G:=sub<GL(3,GF(73))| [1,0,0,0,3,45,0,28,31],[1,0,0,0,28,70,0,42,45],[10,0,0,0,65,57,0,16,8] >;
D18⋊C8 in GAP, Magma, Sage, TeX
D_{18}\rtimes C_8 % in TeX
G:=Group("D18:C8"); // GroupNames label
G:=SmallGroup(288,27);
// by ID
G=gap.SmallGroup(288,27);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^18=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations