metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊3Q8, C36.22D4, (C2×Q8)⋊5D9, C2.9(Q8×D9), (Q8×C18)⋊3C2, C9⋊5(C22⋊Q8), C6.40(S3×Q8), D18⋊C4.6C2, C4⋊Dic9⋊15C2, (C2×C4).19D18, C18.57(C2×D4), C3.(D6⋊3Q8), (C6×Q8).16S3, C18.17(C2×Q8), Dic9⋊C4⋊16C2, (C2×C12).219D6, C4.18(C9⋊D4), C18.36(C4○D4), C12.19(C3⋊D4), (C2×C36).63C22, (C2×C18).58C23, C2.8(Q8⋊3D9), C6.44(Q8⋊3S3), C22.64(C22×D9), (C2×Dic9).18C22, (C22×D9).27C22, (C2×C4×D9).4C2, C2.21(C2×C9⋊D4), C6.105(C2×C3⋊D4), (C2×C6).215(C22×S3), 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 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×5], Q8 [×2], C23, C9, Dic3 [×3], C12 [×2], C12 [×2], D6 [×4], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, D9 [×2], C18 [×3], C4×S3 [×2], C2×Dic3 [×3], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3, C22⋊Q8, Dic9 [×3], C36 [×2], C36 [×2], D18 [×2], D18 [×2], C2×C18, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], S3×C2×C4, C6×Q8, C4×D9 [×2], C2×Dic9, C2×Dic9 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×2], C22×D9, D6⋊3Q8, Dic9⋊C4 [×2], C4⋊Dic9, D18⋊C4 [×2], C2×C4×D9, Q8×C18, D18⋊3Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, D9, C3⋊D4 [×2], C22×S3, C22⋊Q8, D18 [×3], S3×Q8, Q8⋊3S3, C2×C3⋊D4, C9⋊D4 [×2], C22×D9, D6⋊3Q8, Q8×D9, Q8⋊3D9, C2×C9⋊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 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 36)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(73 75)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 92)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(109 117)(110 116)(111 115)(112 114)(118 126)(119 125)(120 124)(121 123)(127 131)(128 130)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)
(1 134 101 88)(2 135 102 89)(3 136 103 90)(4 137 104 73)(5 138 105 74)(6 139 106 75)(7 140 107 76)(8 141 108 77)(9 142 91 78)(10 143 92 79)(11 144 93 80)(12 127 94 81)(13 128 95 82)(14 129 96 83)(15 130 97 84)(16 131 98 85)(17 132 99 86)(18 133 100 87)(19 61 47 110)(20 62 48 111)(21 63 49 112)(22 64 50 113)(23 65 51 114)(24 66 52 115)(25 67 53 116)(26 68 54 117)(27 69 37 118)(28 70 38 119)(29 71 39 120)(30 72 40 121)(31 55 41 122)(32 56 42 123)(33 57 43 124)(34 58 44 125)(35 59 45 126)(36 60 46 109)
(1 27 101 37)(2 28 102 38)(3 29 103 39)(4 30 104 40)(5 31 105 41)(6 32 106 42)(7 33 107 43)(8 34 108 44)(9 35 91 45)(10 36 92 46)(11 19 93 47)(12 20 94 48)(13 21 95 49)(14 22 96 50)(15 23 97 51)(16 24 98 52)(17 25 99 53)(18 26 100 54)(55 138 122 74)(56 139 123 75)(57 140 124 76)(58 141 125 77)(59 142 126 78)(60 143 109 79)(61 144 110 80)(62 127 111 81)(63 128 112 82)(64 129 113 83)(65 130 114 84)(66 131 115 85)(67 132 116 86)(68 133 117 87)(69 134 118 88)(70 135 119 89)(71 136 120 90)(72 137 121 73)
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)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,36)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(73,75)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,92)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(109,117)(110,116)(111,115)(112,114)(118,126)(119,125)(120,124)(121,123)(127,131)(128,130)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139), (1,134,101,88)(2,135,102,89)(3,136,103,90)(4,137,104,73)(5,138,105,74)(6,139,106,75)(7,140,107,76)(8,141,108,77)(9,142,91,78)(10,143,92,79)(11,144,93,80)(12,127,94,81)(13,128,95,82)(14,129,96,83)(15,130,97,84)(16,131,98,85)(17,132,99,86)(18,133,100,87)(19,61,47,110)(20,62,48,111)(21,63,49,112)(22,64,50,113)(23,65,51,114)(24,66,52,115)(25,67,53,116)(26,68,54,117)(27,69,37,118)(28,70,38,119)(29,71,39,120)(30,72,40,121)(31,55,41,122)(32,56,42,123)(33,57,43,124)(34,58,44,125)(35,59,45,126)(36,60,46,109), (1,27,101,37)(2,28,102,38)(3,29,103,39)(4,30,104,40)(5,31,105,41)(6,32,106,42)(7,33,107,43)(8,34,108,44)(9,35,91,45)(10,36,92,46)(11,19,93,47)(12,20,94,48)(13,21,95,49)(14,22,96,50)(15,23,97,51)(16,24,98,52)(17,25,99,53)(18,26,100,54)(55,138,122,74)(56,139,123,75)(57,140,124,76)(58,141,125,77)(59,142,126,78)(60,143,109,79)(61,144,110,80)(62,127,111,81)(63,128,112,82)(64,129,113,83)(65,130,114,84)(66,131,115,85)(67,132,116,86)(68,133,117,87)(69,134,118,88)(70,135,119,89)(71,136,120,90)(72,137,121,73)>;
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)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,36)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(73,75)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,92)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(109,117)(110,116)(111,115)(112,114)(118,126)(119,125)(120,124)(121,123)(127,131)(128,130)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139), (1,134,101,88)(2,135,102,89)(3,136,103,90)(4,137,104,73)(5,138,105,74)(6,139,106,75)(7,140,107,76)(8,141,108,77)(9,142,91,78)(10,143,92,79)(11,144,93,80)(12,127,94,81)(13,128,95,82)(14,129,96,83)(15,130,97,84)(16,131,98,85)(17,132,99,86)(18,133,100,87)(19,61,47,110)(20,62,48,111)(21,63,49,112)(22,64,50,113)(23,65,51,114)(24,66,52,115)(25,67,53,116)(26,68,54,117)(27,69,37,118)(28,70,38,119)(29,71,39,120)(30,72,40,121)(31,55,41,122)(32,56,42,123)(33,57,43,124)(34,58,44,125)(35,59,45,126)(36,60,46,109), (1,27,101,37)(2,28,102,38)(3,29,103,39)(4,30,104,40)(5,31,105,41)(6,32,106,42)(7,33,107,43)(8,34,108,44)(9,35,91,45)(10,36,92,46)(11,19,93,47)(12,20,94,48)(13,21,95,49)(14,22,96,50)(15,23,97,51)(16,24,98,52)(17,25,99,53)(18,26,100,54)(55,138,122,74)(56,139,123,75)(57,140,124,76)(58,141,125,77)(59,142,126,78)(60,143,109,79)(61,144,110,80)(62,127,111,81)(63,128,112,82)(64,129,113,83)(65,130,114,84)(66,131,115,85)(67,132,116,86)(68,133,117,87)(69,134,118,88)(70,135,119,89)(71,136,120,90)(72,137,121,73) );
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),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(35,36),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(73,75),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(91,92),(93,108),(94,107),(95,106),(96,105),(97,104),(98,103),(99,102),(100,101),(109,117),(110,116),(111,115),(112,114),(118,126),(119,125),(120,124),(121,123),(127,131),(128,130),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139)], [(1,134,101,88),(2,135,102,89),(3,136,103,90),(4,137,104,73),(5,138,105,74),(6,139,106,75),(7,140,107,76),(8,141,108,77),(9,142,91,78),(10,143,92,79),(11,144,93,80),(12,127,94,81),(13,128,95,82),(14,129,96,83),(15,130,97,84),(16,131,98,85),(17,132,99,86),(18,133,100,87),(19,61,47,110),(20,62,48,111),(21,63,49,112),(22,64,50,113),(23,65,51,114),(24,66,52,115),(25,67,53,116),(26,68,54,117),(27,69,37,118),(28,70,38,119),(29,71,39,120),(30,72,40,121),(31,55,41,122),(32,56,42,123),(33,57,43,124),(34,58,44,125),(35,59,45,126),(36,60,46,109)], [(1,27,101,37),(2,28,102,38),(3,29,103,39),(4,30,104,40),(5,31,105,41),(6,32,106,42),(7,33,107,43),(8,34,108,44),(9,35,91,45),(10,36,92,46),(11,19,93,47),(12,20,94,48),(13,21,95,49),(14,22,96,50),(15,23,97,51),(16,24,98,52),(17,25,99,53),(18,26,100,54),(55,138,122,74),(56,139,123,75),(57,140,124,76),(58,141,125,77),(59,142,126,78),(60,143,109,79),(61,144,110,80),(62,127,111,81),(63,128,112,82),(64,129,113,83),(65,130,114,84),(66,131,115,85),(67,132,116,86),(68,133,117,87),(69,134,118,88),(70,135,119,89),(71,136,120,90),(72,137,121,73)])
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 | C4○D4 | D9 | C3⋊D4 | D18 | C9⋊D4 | S3×Q8 | Q8⋊3S3 | Q8×D9 | Q8⋊3D9 |
| kernel | D18⋊3Q8 | Dic9⋊C4 | C4⋊Dic9 | D18⋊C4 | C2×C4×D9 | Q8×C18 | C6×Q8 | C36 | D18 | C2×C12 | C18 | C2×Q8 | C12 | C2×C4 | 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