metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D18, C36.50D4, Q8.13D18, C36.17C23, Dic18.11C22, D4.D9⋊6C2, C4○D4.4D9, C9⋊Q16⋊6C2, (C2×C18).8D4, C9⋊C8.4C22, (C3×D4).32D6, (C2×C12).68D6, (C2×C4).20D18, C18.59(C2×D4), C9⋊5(C8.C22), C4.Dic9⋊9C2, (C3×Q8).56D6, C4.24(C9⋊D4), C3.(Q8.14D6), (D4×C9).8C22, C4.17(C22×D9), (C2×Dic18)⋊11C2, (Q8×C9).8C22, (C2×C36).45C22, C12.56(C22×S3), C22.5(C9⋊D4), C12.112(C3⋊D4), (C9×C4○D4).3C2, C2.23(C2×C9⋊D4), (C3×C4○D4).11S3, (C2×C6).7(C3⋊D4), C6.107(C2×C3⋊D4), SmallGroup(288,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D18
G = < a,b,c,d | a4=b2=c18=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 316 in 90 conjugacy classes, 38 normal (30 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C9, Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C18, C18 [×2], C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, Dic9 [×2], C36 [×2], C36, C2×C18, C2×C18, C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C2×Dic6, C3×C4○D4, C9⋊C8 [×2], Dic18 [×2], Dic18, C2×Dic9, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q8.14D6, C4.Dic9, D4.D9 [×2], C9⋊Q16 [×2], C2×Dic18, C9×C4○D4, D4.D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C8.C22, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, Q8.14D6, C2×C9⋊D4, D4.D18
►On 144 points
Generators in S144
(1 102 45 71)(2 103 46 72)(3 104 47 55)(4 105 48 56)(5 106 49 57)(6 107 50 58)(7 108 51 59)(8 91 52 60)(9 92 53 61)(10 93 54 62)(11 94 37 63)(12 95 38 64)(13 96 39 65)(14 97 40 66)(15 98 41 67)(16 99 42 68)(17 100 43 69)(18 101 44 70)(19 140 80 111)(20 141 81 112)(21 142 82 113)(22 143 83 114)(23 144 84 115)(24 127 85 116)(25 128 86 117)(26 129 87 118)(27 130 88 119)(28 131 89 120)(29 132 90 121)(30 133 73 122)(31 134 74 123)(32 135 75 124)(33 136 76 125)(34 137 77 126)(35 138 78 109)(36 139 79 110)
(1 71)(2 103)(3 55)(4 105)(5 57)(6 107)(7 59)(8 91)(9 61)(10 93)(11 63)(12 95)(13 65)(14 97)(15 67)(16 99)(17 69)(18 101)(19 80)(21 82)(23 84)(25 86)(27 88)(29 90)(31 74)(33 76)(35 78)(37 94)(38 64)(39 96)(40 66)(41 98)(42 68)(43 100)(44 70)(45 102)(46 72)(47 104)(48 56)(49 106)(50 58)(51 108)(52 60)(53 92)(54 62)(110 139)(112 141)(114 143)(116 127)(118 129)(120 131)(122 133)(124 135)(126 137)
(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 81 45 20)(2 80 46 19)(3 79 47 36)(4 78 48 35)(5 77 49 34)(6 76 50 33)(7 75 51 32)(8 74 52 31)(9 73 53 30)(10 90 54 29)(11 89 37 28)(12 88 38 27)(13 87 39 26)(14 86 40 25)(15 85 41 24)(16 84 42 23)(17 83 43 22)(18 82 44 21)(55 110 104 139)(56 109 105 138)(57 126 106 137)(58 125 107 136)(59 124 108 135)(60 123 91 134)(61 122 92 133)(62 121 93 132)(63 120 94 131)(64 119 95 130)(65 118 96 129)(66 117 97 128)(67 116 98 127)(68 115 99 144)(69 114 100 143)(70 113 101 142)(71 112 102 141)(72 111 103 140)
G:=sub<Sym(144)| (1,102,45,71)(2,103,46,72)(3,104,47,55)(4,105,48,56)(5,106,49,57)(6,107,50,58)(7,108,51,59)(8,91,52,60)(9,92,53,61)(10,93,54,62)(11,94,37,63)(12,95,38,64)(13,96,39,65)(14,97,40,66)(15,98,41,67)(16,99,42,68)(17,100,43,69)(18,101,44,70)(19,140,80,111)(20,141,81,112)(21,142,82,113)(22,143,83,114)(23,144,84,115)(24,127,85,116)(25,128,86,117)(26,129,87,118)(27,130,88,119)(28,131,89,120)(29,132,90,121)(30,133,73,122)(31,134,74,123)(32,135,75,124)(33,136,76,125)(34,137,77,126)(35,138,78,109)(36,139,79,110), (1,71)(2,103)(3,55)(4,105)(5,57)(6,107)(7,59)(8,91)(9,61)(10,93)(11,63)(12,95)(13,65)(14,97)(15,67)(16,99)(17,69)(18,101)(19,80)(21,82)(23,84)(25,86)(27,88)(29,90)(31,74)(33,76)(35,78)(37,94)(38,64)(39,96)(40,66)(41,98)(42,68)(43,100)(44,70)(45,102)(46,72)(47,104)(48,56)(49,106)(50,58)(51,108)(52,60)(53,92)(54,62)(110,139)(112,141)(114,143)(116,127)(118,129)(120,131)(122,133)(124,135)(126,137), (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), (1,81,45,20)(2,80,46,19)(3,79,47,36)(4,78,48,35)(5,77,49,34)(6,76,50,33)(7,75,51,32)(8,74,52,31)(9,73,53,30)(10,90,54,29)(11,89,37,28)(12,88,38,27)(13,87,39,26)(14,86,40,25)(15,85,41,24)(16,84,42,23)(17,83,43,22)(18,82,44,21)(55,110,104,139)(56,109,105,138)(57,126,106,137)(58,125,107,136)(59,124,108,135)(60,123,91,134)(61,122,92,133)(62,121,93,132)(63,120,94,131)(64,119,95,130)(65,118,96,129)(66,117,97,128)(67,116,98,127)(68,115,99,144)(69,114,100,143)(70,113,101,142)(71,112,102,141)(72,111,103,140)>;
G:=Group( (1,102,45,71)(2,103,46,72)(3,104,47,55)(4,105,48,56)(5,106,49,57)(6,107,50,58)(7,108,51,59)(8,91,52,60)(9,92,53,61)(10,93,54,62)(11,94,37,63)(12,95,38,64)(13,96,39,65)(14,97,40,66)(15,98,41,67)(16,99,42,68)(17,100,43,69)(18,101,44,70)(19,140,80,111)(20,141,81,112)(21,142,82,113)(22,143,83,114)(23,144,84,115)(24,127,85,116)(25,128,86,117)(26,129,87,118)(27,130,88,119)(28,131,89,120)(29,132,90,121)(30,133,73,122)(31,134,74,123)(32,135,75,124)(33,136,76,125)(34,137,77,126)(35,138,78,109)(36,139,79,110), (1,71)(2,103)(3,55)(4,105)(5,57)(6,107)(7,59)(8,91)(9,61)(10,93)(11,63)(12,95)(13,65)(14,97)(15,67)(16,99)(17,69)(18,101)(19,80)(21,82)(23,84)(25,86)(27,88)(29,90)(31,74)(33,76)(35,78)(37,94)(38,64)(39,96)(40,66)(41,98)(42,68)(43,100)(44,70)(45,102)(46,72)(47,104)(48,56)(49,106)(50,58)(51,108)(52,60)(53,92)(54,62)(110,139)(112,141)(114,143)(116,127)(118,129)(120,131)(122,133)(124,135)(126,137), (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), (1,81,45,20)(2,80,46,19)(3,79,47,36)(4,78,48,35)(5,77,49,34)(6,76,50,33)(7,75,51,32)(8,74,52,31)(9,73,53,30)(10,90,54,29)(11,89,37,28)(12,88,38,27)(13,87,39,26)(14,86,40,25)(15,85,41,24)(16,84,42,23)(17,83,43,22)(18,82,44,21)(55,110,104,139)(56,109,105,138)(57,126,106,137)(58,125,107,136)(59,124,108,135)(60,123,91,134)(61,122,92,133)(62,121,93,132)(63,120,94,131)(64,119,95,130)(65,118,96,129)(66,117,97,128)(67,116,98,127)(68,115,99,144)(69,114,100,143)(70,113,101,142)(71,112,102,141)(72,111,103,140) );
G=PermutationGroup([(1,102,45,71),(2,103,46,72),(3,104,47,55),(4,105,48,56),(5,106,49,57),(6,107,50,58),(7,108,51,59),(8,91,52,60),(9,92,53,61),(10,93,54,62),(11,94,37,63),(12,95,38,64),(13,96,39,65),(14,97,40,66),(15,98,41,67),(16,99,42,68),(17,100,43,69),(18,101,44,70),(19,140,80,111),(20,141,81,112),(21,142,82,113),(22,143,83,114),(23,144,84,115),(24,127,85,116),(25,128,86,117),(26,129,87,118),(27,130,88,119),(28,131,89,120),(29,132,90,121),(30,133,73,122),(31,134,74,123),(32,135,75,124),(33,136,76,125),(34,137,77,126),(35,138,78,109),(36,139,79,110)], [(1,71),(2,103),(3,55),(4,105),(5,57),(6,107),(7,59),(8,91),(9,61),(10,93),(11,63),(12,95),(13,65),(14,97),(15,67),(16,99),(17,69),(18,101),(19,80),(21,82),(23,84),(25,86),(27,88),(29,90),(31,74),(33,76),(35,78),(37,94),(38,64),(39,96),(40,66),(41,98),(42,68),(43,100),(44,70),(45,102),(46,72),(47,104),(48,56),(49,106),(50,58),(51,108),(52,60),(53,92),(54,62),(110,139),(112,141),(114,143),(116,127),(118,129),(120,131),(122,133),(124,135),(126,137)], [(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)], [(1,81,45,20),(2,80,46,19),(3,79,47,36),(4,78,48,35),(5,77,49,34),(6,76,50,33),(7,75,51,32),(8,74,52,31),(9,73,53,30),(10,90,54,29),(11,89,37,28),(12,88,38,27),(13,87,39,26),(14,86,40,25),(15,85,41,24),(16,84,42,23),(17,83,43,22),(18,82,44,21),(55,110,104,139),(56,109,105,138),(57,126,106,137),(58,125,107,136),(59,124,108,135),(60,123,91,134),(61,122,92,133),(62,121,93,132),(63,120,94,131),(64,119,95,130),(65,118,96,129),(66,117,97,128),(67,116,98,127),(68,115,99,144),(69,114,100,143),(70,113,101,142),(71,112,102,141),(72,111,103,140)])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 36 | 36 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | D6 | D9 | C3⋊D4 | C3⋊D4 | D18 | D18 | D18 | C9⋊D4 | C9⋊D4 | C8.C22 | Q8.14D6 | D4.D18 |
| kernel | D4.D18 | C4.Dic9 | D4.D9 | C9⋊Q16 | C2×Dic18 | C9×C4○D4 | C3×C4○D4 | C36 | C2×C18 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of D4.D18 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 32 | 1 | 41 | 66 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 30 | 42 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 41 | 72 | 32 | 7 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 31 | 28 | 0 | 0 | 0 | 0 |
| 45 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 68 | 66 | 71 | 49 |
| 0 | 0 | 63 | 59 | 17 | 25 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 49 | 5 | 35 | 19 |
| 54 | 25 | 0 | 0 | 0 | 0 |
| 44 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 6 | 5 |
| 0 | 0 | 63 | 65 | 27 | 28 |
| 0 | 0 | 6 | 0 | 67 | 68 |
| 0 | 0 | 39 | 40 | 0 | 12 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,72,30,0,0,0,1,0,42,0,0,1,41,0,0,0,0,0,66,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,41,72,0,0,0,0,72,0,0,0,0,72,32,0,0,0,0,0,7,0,1],[31,45,0,0,0,0,28,3,0,0,0,0,0,0,68,63,7,49,0,0,66,59,7,5,0,0,71,17,0,35,0,0,49,25,0,19],[54,44,0,0,0,0,25,19,0,0,0,0,0,0,2,63,6,39,0,0,9,65,0,40,0,0,6,27,67,0,0,0,5,28,68,12] >;
D4.D18 in GAP, Magma, Sage, TeX
D_4.D_{18} % in TeX
G:=Group("D4.D18"); // GroupNames label
G:=SmallGroup(288,159);
// by ID
G=gap.SmallGroup(288,159);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,675,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations