metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊1Q8, Dic9.7D4, C4⋊C4⋊4D9, C2.6(Q8×D9), C6.88(S3×D4), C2.14(D4×D9), C9⋊2(C22⋊Q8), C6.36(S3×Q8), D18⋊C4.5C2, (C2×C4).34D18, C18.26(C2×D4), C3.(D6⋊Q8), C18.13(C2×Q8), Dic9⋊C4⋊12C2, (C2×Dic18)⋊4C2, (C2×C12).214D6, C18.13(C4○D4), C6.83(C4○D12), (C2×C36).12C22, (C2×C18).37C23, C2.15(D36⋊5C2), C22.51(C22×D9), (C2×Dic9).10C22, (C22×D9).21C22, (C9×C4⋊C4)⋊7C2, (C2×C4×D9).9C2, (C3×C4⋊C4).14S3, (C2×C6).194(C22×S3), SmallGroup(288,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18⋊Q8
G = < a,b,c,d | a18=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=a7b, dbd-1=a16b, dcd-1=c-1 >
Subgroups: 500 in 111 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, Q8, C23, C9, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, D9, C18, Dic6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊Q8, Dic9, Dic9, C36, D18, D18, C2×C18, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, Dic18, C4×D9, C2×Dic9, C2×C36, C22×D9, D6⋊Q8, Dic9⋊C4, D18⋊C4, C9×C4⋊C4, C2×Dic18, C2×C4×D9, D18⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D9, C22×S3, C22⋊Q8, D18, C4○D12, S3×D4, S3×Q8, C22×D9, D6⋊Q8, D36⋊5C2, D4×D9, Q8×D9, D18⋊Q8
►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 21)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(51 54)(52 53)(55 59)(56 58)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)(73 83)(74 82)(75 81)(76 80)(77 79)(84 90)(85 89)(86 88)(91 101)(92 100)(93 99)(94 98)(95 97)(102 108)(103 107)(104 106)(109 116)(110 115)(111 114)(112 113)(117 126)(118 125)(119 124)(120 123)(121 122)(127 134)(128 133)(129 132)(130 131)(135 144)(136 143)(137 142)(138 141)(139 140)
(1 71 113 34)(2 70 114 33)(3 69 115 32)(4 68 116 31)(5 67 117 30)(6 66 118 29)(7 65 119 28)(8 64 120 27)(9 63 121 26)(10 62 122 25)(11 61 123 24)(12 60 124 23)(13 59 125 22)(14 58 126 21)(15 57 109 20)(16 56 110 19)(17 55 111 36)(18 72 112 35)(37 81 133 99)(38 80 134 98)(39 79 135 97)(40 78 136 96)(41 77 137 95)(42 76 138 94)(43 75 139 93)(44 74 140 92)(45 73 141 91)(46 90 142 108)(47 89 143 107)(48 88 144 106)(49 87 127 105)(50 86 128 104)(51 85 129 103)(52 84 130 102)(53 83 131 101)(54 82 132 100)
(1 131 113 53)(2 130 114 52)(3 129 115 51)(4 128 116 50)(5 127 117 49)(6 144 118 48)(7 143 119 47)(8 142 120 46)(9 141 121 45)(10 140 122 44)(11 139 123 43)(12 138 124 42)(13 137 125 41)(14 136 126 40)(15 135 109 39)(16 134 110 38)(17 133 111 37)(18 132 112 54)(19 98 56 80)(20 97 57 79)(21 96 58 78)(22 95 59 77)(23 94 60 76)(24 93 61 75)(25 92 62 74)(26 91 63 73)(27 108 64 90)(28 107 65 89)(29 106 66 88)(30 105 67 87)(31 104 68 86)(32 103 69 85)(33 102 70 84)(34 101 71 83)(35 100 72 82)(36 99 55 81)
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)(19,21)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(51,54)(52,53)(55,59)(56,58)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(73,83)(74,82)(75,81)(76,80)(77,79)(84,90)(85,89)(86,88)(91,101)(92,100)(93,99)(94,98)(95,97)(102,108)(103,107)(104,106)(109,116)(110,115)(111,114)(112,113)(117,126)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(135,144)(136,143)(137,142)(138,141)(139,140), (1,71,113,34)(2,70,114,33)(3,69,115,32)(4,68,116,31)(5,67,117,30)(6,66,118,29)(7,65,119,28)(8,64,120,27)(9,63,121,26)(10,62,122,25)(11,61,123,24)(12,60,124,23)(13,59,125,22)(14,58,126,21)(15,57,109,20)(16,56,110,19)(17,55,111,36)(18,72,112,35)(37,81,133,99)(38,80,134,98)(39,79,135,97)(40,78,136,96)(41,77,137,95)(42,76,138,94)(43,75,139,93)(44,74,140,92)(45,73,141,91)(46,90,142,108)(47,89,143,107)(48,88,144,106)(49,87,127,105)(50,86,128,104)(51,85,129,103)(52,84,130,102)(53,83,131,101)(54,82,132,100), (1,131,113,53)(2,130,114,52)(3,129,115,51)(4,128,116,50)(5,127,117,49)(6,144,118,48)(7,143,119,47)(8,142,120,46)(9,141,121,45)(10,140,122,44)(11,139,123,43)(12,138,124,42)(13,137,125,41)(14,136,126,40)(15,135,109,39)(16,134,110,38)(17,133,111,37)(18,132,112,54)(19,98,56,80)(20,97,57,79)(21,96,58,78)(22,95,59,77)(23,94,60,76)(24,93,61,75)(25,92,62,74)(26,91,63,73)(27,108,64,90)(28,107,65,89)(29,106,66,88)(30,105,67,87)(31,104,68,86)(32,103,69,85)(33,102,70,84)(34,101,71,83)(35,100,72,82)(36,99,55,81)>;
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)(19,21)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(51,54)(52,53)(55,59)(56,58)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(73,83)(74,82)(75,81)(76,80)(77,79)(84,90)(85,89)(86,88)(91,101)(92,100)(93,99)(94,98)(95,97)(102,108)(103,107)(104,106)(109,116)(110,115)(111,114)(112,113)(117,126)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(135,144)(136,143)(137,142)(138,141)(139,140), (1,71,113,34)(2,70,114,33)(3,69,115,32)(4,68,116,31)(5,67,117,30)(6,66,118,29)(7,65,119,28)(8,64,120,27)(9,63,121,26)(10,62,122,25)(11,61,123,24)(12,60,124,23)(13,59,125,22)(14,58,126,21)(15,57,109,20)(16,56,110,19)(17,55,111,36)(18,72,112,35)(37,81,133,99)(38,80,134,98)(39,79,135,97)(40,78,136,96)(41,77,137,95)(42,76,138,94)(43,75,139,93)(44,74,140,92)(45,73,141,91)(46,90,142,108)(47,89,143,107)(48,88,144,106)(49,87,127,105)(50,86,128,104)(51,85,129,103)(52,84,130,102)(53,83,131,101)(54,82,132,100), (1,131,113,53)(2,130,114,52)(3,129,115,51)(4,128,116,50)(5,127,117,49)(6,144,118,48)(7,143,119,47)(8,142,120,46)(9,141,121,45)(10,140,122,44)(11,139,123,43)(12,138,124,42)(13,137,125,41)(14,136,126,40)(15,135,109,39)(16,134,110,38)(17,133,111,37)(18,132,112,54)(19,98,56,80)(20,97,57,79)(21,96,58,78)(22,95,59,77)(23,94,60,76)(24,93,61,75)(25,92,62,74)(26,91,63,73)(27,108,64,90)(28,107,65,89)(29,106,66,88)(30,105,67,87)(31,104,68,86)(32,103,69,85)(33,102,70,84)(34,101,71,83)(35,100,72,82)(36,99,55,81) );
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),(19,21),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(51,54),(52,53),(55,59),(56,58),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67),(73,83),(74,82),(75,81),(76,80),(77,79),(84,90),(85,89),(86,88),(91,101),(92,100),(93,99),(94,98),(95,97),(102,108),(103,107),(104,106),(109,116),(110,115),(111,114),(112,113),(117,126),(118,125),(119,124),(120,123),(121,122),(127,134),(128,133),(129,132),(130,131),(135,144),(136,143),(137,142),(138,141),(139,140)], [(1,71,113,34),(2,70,114,33),(3,69,115,32),(4,68,116,31),(5,67,117,30),(6,66,118,29),(7,65,119,28),(8,64,120,27),(9,63,121,26),(10,62,122,25),(11,61,123,24),(12,60,124,23),(13,59,125,22),(14,58,126,21),(15,57,109,20),(16,56,110,19),(17,55,111,36),(18,72,112,35),(37,81,133,99),(38,80,134,98),(39,79,135,97),(40,78,136,96),(41,77,137,95),(42,76,138,94),(43,75,139,93),(44,74,140,92),(45,73,141,91),(46,90,142,108),(47,89,143,107),(48,88,144,106),(49,87,127,105),(50,86,128,104),(51,85,129,103),(52,84,130,102),(53,83,131,101),(54,82,132,100)], [(1,131,113,53),(2,130,114,52),(3,129,115,51),(4,128,116,50),(5,127,117,49),(6,144,118,48),(7,143,119,47),(8,142,120,46),(9,141,121,45),(10,140,122,44),(11,139,123,43),(12,138,124,42),(13,137,125,41),(14,136,126,40),(15,135,109,39),(16,134,110,38),(17,133,111,37),(18,132,112,54),(19,98,56,80),(20,97,57,79),(21,96,58,78),(22,95,59,77),(23,94,60,76),(24,93,61,75),(25,92,62,74),(26,91,63,73),(27,108,64,90),(28,107,65,89),(29,106,66,88),(30,105,67,87),(31,104,68,86),(32,103,69,85),(33,102,70,84),(34,101,71,83),(35,100,72,82),(36,99,55,81)]])
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 | D18 | C4○D12 | D36⋊5C2 | S3×D4 | S3×Q8 | D4×D9 | Q8×D9 |
| kernel | D18⋊Q8 | Dic9⋊C4 | D18⋊C4 | C9×C4⋊C4 | C2×Dic18 | C2×C4×D9 | C3×C4⋊C4 | Dic9 | D18 | C2×C12 | C18 | C4⋊C4 | C2×C4 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 9 | 4 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of D18⋊Q8 ►in GL4(𝔽37) generated by
| 17 | 31 | 0 | 0 |
| 6 | 11 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 6 | 11 | 0 | 0 |
| 17 | 31 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 13 | 1 |
| 23 | 7 | 0 | 0 |
| 30 | 14 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 36 |
| 0 | 31 | 0 | 0 |
| 31 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [17,6,0,0,31,11,0,0,0,0,36,0,0,0,0,36],[6,17,0,0,11,31,0,0,0,0,36,13,0,0,0,1],[23,30,0,0,7,14,0,0,0,0,1,0,0,0,3,36],[0,31,0,0,31,0,0,0,0,0,1,0,0,0,0,1] >;
D18⋊Q8 in GAP, Magma, Sage, TeX
D_{18}\rtimes Q_8 % in TeX
G:=Group("D18:Q8"); // GroupNames label
G:=SmallGroup(288,106);
// by ID
G=gap.SmallGroup(288,106);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^7*b,d*b*d^-1=a^16*b,d*c*d^-1=c^-1>;
// generators/relations