metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊1D4, Dic9⋊2D4, C23.10D18, C2.9(D4×D9), (C2×D36)⋊3C2, C9⋊1(C4⋊D4), C22⋊C4⋊4D9, C6.81(S3×D4), D18⋊C4⋊10C2, Dic9⋊C4⋊5C2, C3.(Dic3⋊D4), (C2×C4).27D18, C18.20(C2×D4), C18.9(C4○D4), (C2×C12).178D6, (C2×C36).4C22, (C22×C6).44D6, C6.79(C4○D12), (C2×C18).25C23, C2.11(D36⋊5C2), (C2×Dic9).6C22, (C22×D9).4C22, C22.43(C22×D9), (C22×C18).14C22, (C2×C4×D9)⋊11C2, (C2×C9⋊D4)⋊2C2, (C9×C22⋊C4)⋊6C2, (C3×C22⋊C4).8S3, (C2×C6).182(C22×S3), SmallGroup(288,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18⋊D4
G = < a,b,c,d | a18=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a16b, dbd=a7b, dcd=c-1 >
Subgroups: 732 in 141 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D9, C18, C18, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4⋊D4, Dic9, Dic9, C36, D18, D18, C2×C18, C2×C18, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, C22×D9, C22×C18, Dic3⋊D4, Dic9⋊C4, D18⋊C4, C9×C22⋊C4, C2×C4×D9, C2×D36, C2×C9⋊D4, D18⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C22×S3, C4⋊D4, D18, C4○D12, S3×D4, C22×D9, Dic3⋊D4, D36⋊5C2, D4×D9, D18⋊D4
►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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(37 107)(38 106)(39 105)(40 104)(41 103)(42 102)(43 101)(44 100)(45 99)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 108)(55 130)(56 129)(57 128)(58 127)(59 144)(60 143)(61 142)(62 141)(63 140)(64 139)(65 138)(66 137)(67 136)(68 135)(69 134)(70 133)(71 132)(72 131)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 110)(81 109)(82 126)(83 125)(84 124)(85 123)(86 122)(87 121)(88 120)(89 119)(90 118)
(1 84 138 38)(2 83 139 37)(3 82 140 54)(4 81 141 53)(5 80 142 52)(6 79 143 51)(7 78 144 50)(8 77 127 49)(9 76 128 48)(10 75 129 47)(11 74 130 46)(12 73 131 45)(13 90 132 44)(14 89 133 43)(15 88 134 42)(16 87 135 41)(17 86 136 40)(18 85 137 39)(19 117 56 99)(20 116 57 98)(21 115 58 97)(22 114 59 96)(23 113 60 95)(24 112 61 94)(25 111 62 93)(26 110 63 92)(27 109 64 91)(28 126 65 108)(29 125 66 107)(30 124 67 106)(31 123 68 105)(32 122 69 104)(33 121 70 103)(34 120 71 102)(35 119 72 101)(36 118 55 100)
(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(31 36)(32 35)(33 34)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 90)(51 89)(52 88)(53 87)(54 86)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(69 72)(70 71)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(127 131)(128 130)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(37,107)(38,106)(39,105)(40,104)(41,103)(42,102)(43,101)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,108)(55,130)(56,129)(57,128)(58,127)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,132)(72,131)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,126)(83,125)(84,124)(85,123)(86,122)(87,121)(88,120)(89,119)(90,118), (1,84,138,38)(2,83,139,37)(3,82,140,54)(4,81,141,53)(5,80,142,52)(6,79,143,51)(7,78,144,50)(8,77,127,49)(9,76,128,48)(10,75,129,47)(11,74,130,46)(12,73,131,45)(13,90,132,44)(14,89,133,43)(15,88,134,42)(16,87,135,41)(17,86,136,40)(18,85,137,39)(19,117,56,99)(20,116,57,98)(21,115,58,97)(22,114,59,96)(23,113,60,95)(24,112,61,94)(25,111,62,93)(26,110,63,92)(27,109,64,91)(28,126,65,108)(29,125,66,107)(30,124,67,106)(31,123,68,105)(32,122,69,104)(33,121,70,103)(34,120,71,102)(35,119,72,101)(36,118,55,100), (2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(31,36)(32,35)(33,34)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,90)(51,89)(52,88)(53,87)(54,86)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,72)(70,71)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(127,131)(128,130)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(37,107)(38,106)(39,105)(40,104)(41,103)(42,102)(43,101)(44,100)(45,99)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,108)(55,130)(56,129)(57,128)(58,127)(59,144)(60,143)(61,142)(62,141)(63,140)(64,139)(65,138)(66,137)(67,136)(68,135)(69,134)(70,133)(71,132)(72,131)(73,117)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,126)(83,125)(84,124)(85,123)(86,122)(87,121)(88,120)(89,119)(90,118), (1,84,138,38)(2,83,139,37)(3,82,140,54)(4,81,141,53)(5,80,142,52)(6,79,143,51)(7,78,144,50)(8,77,127,49)(9,76,128,48)(10,75,129,47)(11,74,130,46)(12,73,131,45)(13,90,132,44)(14,89,133,43)(15,88,134,42)(16,87,135,41)(17,86,136,40)(18,85,137,39)(19,117,56,99)(20,116,57,98)(21,115,58,97)(22,114,59,96)(23,113,60,95)(24,112,61,94)(25,111,62,93)(26,110,63,92)(27,109,64,91)(28,126,65,108)(29,125,66,107)(30,124,67,106)(31,123,68,105)(32,122,69,104)(33,121,70,103)(34,120,71,102)(35,119,72,101)(36,118,55,100), (2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(31,36)(32,35)(33,34)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,90)(51,89)(52,88)(53,87)(54,86)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,72)(70,71)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,126)(98,125)(99,124)(100,123)(101,122)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)(108,115)(127,131)(128,130)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(37,107),(38,106),(39,105),(40,104),(41,103),(42,102),(43,101),(44,100),(45,99),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,108),(55,130),(56,129),(57,128),(58,127),(59,144),(60,143),(61,142),(62,141),(63,140),(64,139),(65,138),(66,137),(67,136),(68,135),(69,134),(70,133),(71,132),(72,131),(73,117),(74,116),(75,115),(76,114),(77,113),(78,112),(79,111),(80,110),(81,109),(82,126),(83,125),(84,124),(85,123),(86,122),(87,121),(88,120),(89,119),(90,118)], [(1,84,138,38),(2,83,139,37),(3,82,140,54),(4,81,141,53),(5,80,142,52),(6,79,143,51),(7,78,144,50),(8,77,127,49),(9,76,128,48),(10,75,129,47),(11,74,130,46),(12,73,131,45),(13,90,132,44),(14,89,133,43),(15,88,134,42),(16,87,135,41),(17,86,136,40),(18,85,137,39),(19,117,56,99),(20,116,57,98),(21,115,58,97),(22,114,59,96),(23,113,60,95),(24,112,61,94),(25,111,62,93),(26,110,63,92),(27,109,64,91),(28,126,65,108),(29,125,66,107),(30,124,67,106),(31,123,68,105),(32,122,69,104),(33,121,70,103),(34,120,71,102),(35,119,72,101),(36,118,55,100)], [(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(31,36),(32,35),(33,34),(37,85),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,90),(51,89),(52,88),(53,87),(54,86),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(69,72),(70,71),(91,114),(92,113),(93,112),(94,111),(95,110),(96,109),(97,126),(98,125),(99,124),(100,123),(101,122),(102,121),(103,120),(104,119),(105,118),(106,117),(107,116),(108,115),(127,131),(128,130),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 18 | 18 | 36 | 2 | 2 | 2 | 4 | 18 | 18 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | D9 | D18 | D18 | C4○D12 | D36⋊5C2 | S3×D4 | D4×D9 |
| kernel | D18⋊D4 | Dic9⋊C4 | D18⋊C4 | C9×C22⋊C4 | C2×C4×D9 | C2×D36 | C2×C9⋊D4 | C3×C22⋊C4 | Dic9 | D18 | C2×C12 | C22×C6 | C18 | C22⋊C4 | C2×C4 | C23 | C6 | C2 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 3 | 6 | 3 | 4 | 12 | 2 | 6 |
Matrix representation of D18⋊D4 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 6 | 17 |
| 0 | 0 | 20 | 26 |
| 36 | 3 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 12 |
| 0 | 0 | 8 | 33 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 2 | 13 |
| 0 | 0 | 11 | 35 |
| 1 | 0 | 0 | 0 |
| 13 | 36 | 0 | 0 |
| 0 | 0 | 20 | 26 |
| 0 | 0 | 6 | 17 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,6,20,0,0,17,26],[36,0,0,0,3,1,0,0,0,0,4,8,0,0,12,33],[36,0,0,0,0,36,0,0,0,0,2,11,0,0,13,35],[1,13,0,0,0,36,0,0,0,0,20,6,0,0,26,17] >;
D18⋊D4 in GAP, Magma, Sage, TeX
D_{18}\rtimes D_4 % in TeX
G:=Group("D18:D4"); // GroupNames label
G:=SmallGroup(288,94);
// by ID
G=gap.SmallGroup(288,94);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^16*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations