direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C18, C24⋊6C18, C36⋊4C23, C18.16C24, C4⋊(C22×C18), C6.80(C6×D4), C23⋊4(C2×C18), (C2×C18)⋊2C23, (C23×C18)⋊2C2, (C22×C4)⋊7C18, (C6×D4).27C6, (C2×C36)⋊15C22, (C22×C36)⋊12C2, C2.1(C23×C18), (C23×C6).10C6, C6.16(C23×C6), C12.50(C22×C6), (C22×C12).31C6, C22⋊2(C22×C18), (C22×C18)⋊6C22, C3.(D4×C2×C6), (D4×C2×C6).2C3, (C2×C4)⋊4(C2×C18), (C2×C6).69(C3×D4), (C3×D4).19(C2×C6), (C2×C6).7(C22×C6), (C2×C12).154(C2×C6), (C22×C6).49(C2×C6), SmallGroup(288,368)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C2×C18
G = < a,b,c,d | a2=b18=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 474 in 354 conjugacy classes, 234 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, D4, C23, C23, C23, C9, C12, C2×C6, C2×C6, C22×C4, C2×D4, C24, C18, C18, C18, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C22×D4, C36, C2×C18, C2×C18, C22×C12, C6×D4, C23×C6, C2×C36, D4×C9, C22×C18, C22×C18, C22×C18, D4×C2×C6, C22×C36, D4×C18, C23×C18, D4×C2×C18
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C24, C18, C3×D4, C22×C6, C22×D4, C2×C18, C6×D4, C23×C6, D4×C9, C22×C18, D4×C2×C6, D4×C18, C23×C18, D4×C2×C18
►On 144 points
Generators in S144
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 141)(8 142)(9 143)(10 144)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 86)(20 87)(21 88)(22 89)(23 90)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(91 122)(92 123)(93 124)(94 125)(95 126)(96 109)(97 110)(98 111)(99 112)(100 113)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 121)
(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)
(1 113 83 41)(2 114 84 42)(3 115 85 43)(4 116 86 44)(5 117 87 45)(6 118 88 46)(7 119 89 47)(8 120 90 48)(9 121 73 49)(10 122 74 50)(11 123 75 51)(12 124 76 52)(13 125 77 53)(14 126 78 54)(15 109 79 37)(16 110 80 38)(17 111 81 39)(18 112 82 40)(19 68 138 103)(20 69 139 104)(21 70 140 105)(22 71 141 106)(23 72 142 107)(24 55 143 108)(25 56 144 91)(26 57 127 92)(27 58 128 93)(28 59 129 94)(29 60 130 95)(30 61 131 96)(31 62 132 97)(32 63 133 98)(33 64 134 99)(34 65 135 100)(35 66 136 101)(36 67 137 102)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 70)(38 71)(39 72)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(73 134)(74 135)(75 136)(76 137)(77 138)(78 139)(79 140)(80 141)(81 142)(82 143)(83 144)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 109)(106 110)(107 111)(108 112)
G:=sub<Sym(144)| (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,141)(8,142)(9,143)(10,144)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,86)(20,87)(21,88)(22,89)(23,90)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(91,122)(92,123)(93,124)(94,125)(95,126)(96,109)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121), (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), (1,113,83,41)(2,114,84,42)(3,115,85,43)(4,116,86,44)(5,117,87,45)(6,118,88,46)(7,119,89,47)(8,120,90,48)(9,121,73,49)(10,122,74,50)(11,123,75,51)(12,124,76,52)(13,125,77,53)(14,126,78,54)(15,109,79,37)(16,110,80,38)(17,111,81,39)(18,112,82,40)(19,68,138,103)(20,69,139,104)(21,70,140,105)(22,71,141,106)(23,72,142,107)(24,55,143,108)(25,56,144,91)(26,57,127,92)(27,58,128,93)(28,59,129,94)(29,60,130,95)(30,61,131,96)(31,62,132,97)(32,63,133,98)(33,64,134,99)(34,65,135,100)(35,66,136,101)(36,67,137,102), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,70)(38,71)(39,72)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(73,134)(74,135)(75,136)(76,137)(77,138)(78,139)(79,140)(80,141)(81,142)(82,143)(83,144)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,109)(106,110)(107,111)(108,112)>;
G:=Group( (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,141)(8,142)(9,143)(10,144)(11,127)(12,128)(13,129)(14,130)(15,131)(16,132)(17,133)(18,134)(19,86)(20,87)(21,88)(22,89)(23,90)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,81)(33,82)(34,83)(35,84)(36,85)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(91,122)(92,123)(93,124)(94,125)(95,126)(96,109)(97,110)(98,111)(99,112)(100,113)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,121), (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), (1,113,83,41)(2,114,84,42)(3,115,85,43)(4,116,86,44)(5,117,87,45)(6,118,88,46)(7,119,89,47)(8,120,90,48)(9,121,73,49)(10,122,74,50)(11,123,75,51)(12,124,76,52)(13,125,77,53)(14,126,78,54)(15,109,79,37)(16,110,80,38)(17,111,81,39)(18,112,82,40)(19,68,138,103)(20,69,139,104)(21,70,140,105)(22,71,141,106)(23,72,142,107)(24,55,143,108)(25,56,144,91)(26,57,127,92)(27,58,128,93)(28,59,129,94)(29,60,130,95)(30,61,131,96)(31,62,132,97)(32,63,133,98)(33,64,134,99)(34,65,135,100)(35,66,136,101)(36,67,137,102), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,70)(38,71)(39,72)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(73,134)(74,135)(75,136)(76,137)(77,138)(78,139)(79,140)(80,141)(81,142)(82,143)(83,144)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,109)(106,110)(107,111)(108,112) );
G=PermutationGroup([[(1,135),(2,136),(3,137),(4,138),(5,139),(6,140),(7,141),(8,142),(9,143),(10,144),(11,127),(12,128),(13,129),(14,130),(15,131),(16,132),(17,133),(18,134),(19,86),(20,87),(21,88),(22,89),(23,90),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,81),(33,82),(34,83),(35,84),(36,85),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(91,122),(92,123),(93,124),(94,125),(95,126),(96,109),(97,110),(98,111),(99,112),(100,113),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,121)], [(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)], [(1,113,83,41),(2,114,84,42),(3,115,85,43),(4,116,86,44),(5,117,87,45),(6,118,88,46),(7,119,89,47),(8,120,90,48),(9,121,73,49),(10,122,74,50),(11,123,75,51),(12,124,76,52),(13,125,77,53),(14,126,78,54),(15,109,79,37),(16,110,80,38),(17,111,81,39),(18,112,82,40),(19,68,138,103),(20,69,139,104),(21,70,140,105),(22,71,141,106),(23,72,142,107),(24,55,143,108),(25,56,144,91),(26,57,127,92),(27,58,128,93),(28,59,129,94),(29,60,130,95),(30,61,131,96),(31,62,132,97),(32,63,133,98),(33,64,134,99),(34,65,135,100),(35,66,136,101),(36,67,137,102)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(37,70),(38,71),(39,72),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(73,134),(74,135),(75,136),(76,137),(77,138),(78,139),(79,140),(80,141),(81,142),(82,143),(83,144),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,109),(106,110),(107,111),(108,112)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6N | 6O | ··· | 6AD | 9A | ··· | 9F | 12A | ··· | 12H | 18A | ··· | 18AP | 18AQ | ··· | 18CL | 36A | ··· | 36X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | D4 | C3×D4 | D4×C9 |
| kernel | D4×C2×C18 | C22×C36 | D4×C18 | C23×C18 | D4×C2×C6 | C22×C12 | C6×D4 | C23×C6 | C22×D4 | C22×C4 | C2×D4 | C24 | C2×C18 | C2×C6 | C22 |
| # reps | 1 | 1 | 12 | 2 | 2 | 2 | 24 | 4 | 6 | 6 | 72 | 12 | 4 | 8 | 24 |
Matrix representation of D4×C2×C18 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 0 | 33 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 2 |
| 0 | 0 | 36 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 36 | 1 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,0,10,0,0,0,0,33,0,0,0,0,33],[36,0,0,0,0,36,0,0,0,0,36,36,0,0,2,1],[36,0,0,0,0,1,0,0,0,0,36,36,0,0,0,1] >;
D4×C2×C18 in GAP, Magma, Sage, TeX
D_4\times C_2\times C_{18} % in TeX
G:=Group("D4xC2xC18"); // GroupNames label
G:=SmallGroup(288,368);
// by ID
G=gap.SmallGroup(288,368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,242]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^18=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations