metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18:3Q8, C36.22D4, (C2xQ8):5D9, C2.9(Q8xD9), (Q8xC18):3C2, C9:5(C22:Q8), C6.40(S3xQ8), D18:C4.6C2, C4:Dic9:15C2, (C2xC4).19D18, C18.57(C2xD4), C3.(D6:3Q8), (C6xQ8).16S3, C18.17(C2xQ8), Dic9:C4:16C2, (C2xC12).219D6, C4.18(C9:D4), C18.36(C4oD4), C12.19(C3:D4), (C2xC36).63C22, (C2xC18).58C23, C2.8(Q8:3D9), C6.44(Q8:3S3), C22.64(C22xD9), (C2xDic9).18C22, (C22xD9).27C22, (C2xC4xD9).4C2, C2.21(C2xC9:D4), C6.105(C2xC3:D4), (C2xC6).215(C22xS3), SmallGroup(288,156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18:3Q8
G = < a,b,c,d | a18=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >
Subgroups: 460 in 111 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2xC4, C2xC4, C2xC4, Q8, C23, C9, Dic3, C12, C12, D6, C2xC6, C22:C4, C4:C4, C22xC4, C2xQ8, D9, C18, C4xS3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C22:Q8, Dic9, C36, C36, D18, D18, C2xC18, Dic3:C4, C4:Dic3, D6:C4, S3xC2xC4, C6xQ8, C4xD9, C2xDic9, C2xDic9, C2xC36, C2xC36, Q8xC9, C22xD9, D6:3Q8, Dic9:C4, C4:Dic9, D18:C4, C2xC4xD9, Q8xC18, D18:3Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, D9, C3:D4, C22xS3, C22:Q8, D18, S3xQ8, Q8:3S3, C2xC3:D4, C9:D4, C22xD9, D6:3Q8, Q8xD9, Q8:3D9, C2xC9:D4, D18:3Q8
►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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 25)(20 24)(21 23)(26 36)(27 35)(28 34)(29 33)(30 32)(37 40)(38 39)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(88 90)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(109 121)(110 120)(111 119)(112 118)(113 117)(114 116)(122 126)(123 125)(127 134)(128 133)(129 132)(130 131)(135 144)(136 143)(137 142)(138 141)(139 140)
(1 105 48 27)(2 106 49 28)(3 107 50 29)(4 108 51 30)(5 91 52 31)(6 92 53 32)(7 93 54 33)(8 94 37 34)(9 95 38 35)(10 96 39 36)(11 97 40 19)(12 98 41 20)(13 99 42 21)(14 100 43 22)(15 101 44 23)(16 102 45 24)(17 103 46 25)(18 104 47 26)(55 73 137 126)(56 74 138 109)(57 75 139 110)(58 76 140 111)(59 77 141 112)(60 78 142 113)(61 79 143 114)(62 80 144 115)(63 81 127 116)(64 82 128 117)(65 83 129 118)(66 84 130 119)(67 85 131 120)(68 86 132 121)(69 87 133 122)(70 88 134 123)(71 89 135 124)(72 90 136 125)
(1 58 48 140)(2 59 49 141)(3 60 50 142)(4 61 51 143)(5 62 52 144)(6 63 53 127)(7 64 54 128)(8 65 37 129)(9 66 38 130)(10 67 39 131)(11 68 40 132)(12 69 41 133)(13 70 42 134)(14 71 43 135)(15 72 44 136)(16 55 45 137)(17 56 46 138)(18 57 47 139)(19 86 97 121)(20 87 98 122)(21 88 99 123)(22 89 100 124)(23 90 101 125)(24 73 102 126)(25 74 103 109)(26 75 104 110)(27 76 105 111)(28 77 106 112)(29 78 107 113)(30 79 108 114)(31 80 91 115)(32 81 92 116)(33 82 93 117)(34 83 94 118)(35 84 95 119)(36 85 96 120)
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,25)(20,24)(21,23)(26,36)(27,35)(28,34)(29,33)(30,32)(37,40)(38,39)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,90)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(109,121)(110,120)(111,119)(112,118)(113,117)(114,116)(122,126)(123,125)(127,134)(128,133)(129,132)(130,131)(135,144)(136,143)(137,142)(138,141)(139,140), (1,105,48,27)(2,106,49,28)(3,107,50,29)(4,108,51,30)(5,91,52,31)(6,92,53,32)(7,93,54,33)(8,94,37,34)(9,95,38,35)(10,96,39,36)(11,97,40,19)(12,98,41,20)(13,99,42,21)(14,100,43,22)(15,101,44,23)(16,102,45,24)(17,103,46,25)(18,104,47,26)(55,73,137,126)(56,74,138,109)(57,75,139,110)(58,76,140,111)(59,77,141,112)(60,78,142,113)(61,79,143,114)(62,80,144,115)(63,81,127,116)(64,82,128,117)(65,83,129,118)(66,84,130,119)(67,85,131,120)(68,86,132,121)(69,87,133,122)(70,88,134,123)(71,89,135,124)(72,90,136,125), (1,58,48,140)(2,59,49,141)(3,60,50,142)(4,61,51,143)(5,62,52,144)(6,63,53,127)(7,64,54,128)(8,65,37,129)(9,66,38,130)(10,67,39,131)(11,68,40,132)(12,69,41,133)(13,70,42,134)(14,71,43,135)(15,72,44,136)(16,55,45,137)(17,56,46,138)(18,57,47,139)(19,86,97,121)(20,87,98,122)(21,88,99,123)(22,89,100,124)(23,90,101,125)(24,73,102,126)(25,74,103,109)(26,75,104,110)(27,76,105,111)(28,77,106,112)(29,78,107,113)(30,79,108,114)(31,80,91,115)(32,81,92,116)(33,82,93,117)(34,83,94,118)(35,84,95,119)(36,85,96,120)>;
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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,25)(20,24)(21,23)(26,36)(27,35)(28,34)(29,33)(30,32)(37,40)(38,39)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81)(88,90)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(109,121)(110,120)(111,119)(112,118)(113,117)(114,116)(122,126)(123,125)(127,134)(128,133)(129,132)(130,131)(135,144)(136,143)(137,142)(138,141)(139,140), (1,105,48,27)(2,106,49,28)(3,107,50,29)(4,108,51,30)(5,91,52,31)(6,92,53,32)(7,93,54,33)(8,94,37,34)(9,95,38,35)(10,96,39,36)(11,97,40,19)(12,98,41,20)(13,99,42,21)(14,100,43,22)(15,101,44,23)(16,102,45,24)(17,103,46,25)(18,104,47,26)(55,73,137,126)(56,74,138,109)(57,75,139,110)(58,76,140,111)(59,77,141,112)(60,78,142,113)(61,79,143,114)(62,80,144,115)(63,81,127,116)(64,82,128,117)(65,83,129,118)(66,84,130,119)(67,85,131,120)(68,86,132,121)(69,87,133,122)(70,88,134,123)(71,89,135,124)(72,90,136,125), (1,58,48,140)(2,59,49,141)(3,60,50,142)(4,61,51,143)(5,62,52,144)(6,63,53,127)(7,64,54,128)(8,65,37,129)(9,66,38,130)(10,67,39,131)(11,68,40,132)(12,69,41,133)(13,70,42,134)(14,71,43,135)(15,72,44,136)(16,55,45,137)(17,56,46,138)(18,57,47,139)(19,86,97,121)(20,87,98,122)(21,88,99,123)(22,89,100,124)(23,90,101,125)(24,73,102,126)(25,74,103,109)(26,75,104,110)(27,76,105,111)(28,77,106,112)(29,78,107,113)(30,79,108,114)(31,80,91,115)(32,81,92,116)(33,82,93,117)(34,83,94,118)(35,84,95,119)(36,85,96,120) );
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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,25),(20,24),(21,23),(26,36),(27,35),(28,34),(29,33),(30,32),(37,40),(38,39),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82),(79,81),(88,90),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(109,121),(110,120),(111,119),(112,118),(113,117),(114,116),(122,126),(123,125),(127,134),(128,133),(129,132),(130,131),(135,144),(136,143),(137,142),(138,141),(139,140)], [(1,105,48,27),(2,106,49,28),(3,107,50,29),(4,108,51,30),(5,91,52,31),(6,92,53,32),(7,93,54,33),(8,94,37,34),(9,95,38,35),(10,96,39,36),(11,97,40,19),(12,98,41,20),(13,99,42,21),(14,100,43,22),(15,101,44,23),(16,102,45,24),(17,103,46,25),(18,104,47,26),(55,73,137,126),(56,74,138,109),(57,75,139,110),(58,76,140,111),(59,77,141,112),(60,78,142,113),(61,79,143,114),(62,80,144,115),(63,81,127,116),(64,82,128,117),(65,83,129,118),(66,84,130,119),(67,85,131,120),(68,86,132,121),(69,87,133,122),(70,88,134,123),(71,89,135,124),(72,90,136,125)], [(1,58,48,140),(2,59,49,141),(3,60,50,142),(4,61,51,143),(5,62,52,144),(6,63,53,127),(7,64,54,128),(8,65,37,129),(9,66,38,130),(10,67,39,131),(11,68,40,132),(12,69,41,133),(13,70,42,134),(14,71,43,135),(15,72,44,136),(16,55,45,137),(17,56,46,138),(18,57,47,139),(19,86,97,121),(20,87,98,122),(21,88,99,123),(22,89,100,124),(23,90,101,125),(24,73,102,126),(25,74,103,109),(26,75,104,110),(27,76,105,111),(28,77,106,112),(29,78,107,113),(30,79,108,114),(31,80,91,115),(32,81,92,116),(33,82,93,117),(34,83,94,118),(35,84,95,119),(36,85,96,120)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | C4oD4 | D9 | C3:D4 | D18 | C9:D4 | S3xQ8 | Q8:3S3 | Q8xD9 | Q8:3D9 |
| kernel | D18:3Q8 | Dic9:C4 | C4:Dic9 | D18:C4 | C2xC4xD9 | Q8xC18 | C6xQ8 | C36 | D18 | C2xC12 | C18 | C2xQ8 | C12 | C2xC4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of D18:3Q8 ►in GL4(F37) generated by
| 6 | 17 | 0 | 0 |
| 20 | 26 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 6 | 17 | 0 | 0 |
| 11 | 31 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 36 | 1 |
| 7 | 14 | 0 | 0 |
| 23 | 30 | 0 | 0 |
| 0 | 0 | 1 | 35 |
| 0 | 0 | 1 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 6 | 31 |
G:=sub<GL(4,GF(37))| [6,20,0,0,17,26,0,0,0,0,36,0,0,0,0,36],[6,11,0,0,17,31,0,0,0,0,36,36,0,0,0,1],[7,23,0,0,14,30,0,0,0,0,1,1,0,0,35,36],[1,0,0,0,0,1,0,0,0,0,6,6,0,0,0,31] >;
D18:3Q8 in GAP, Magma, Sage, TeX
D_{18}\rtimes_3Q_8 % in TeX
G:=Group("D18:3Q8"); // GroupNames label
G:=SmallGroup(288,156);
// by ID
G=gap.SmallGroup(288,156);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations