direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C18, C18.18C24, C36.51C23, (C2×D4)⋊7C18, D4⋊3(C2×C18), Q8⋊4(C2×C18), (C2×Q8)⋊8C18, (D4×C18)⋊16C2, (C22×C4)⋊8C18, (Q8×C18)⋊13C2, (C6×D4).28C6, (C6×Q8).29C6, (C22×C36)⋊13C2, (C2×C36)⋊16C22, (D4×C9)⋊12C22, C2.3(C23×C18), C4.8(C22×C18), C6.18(C23×C6), (C2×C18).6C23, (Q8×C9)⋊11C22, (C22×C12).33C6, C23.14(C2×C18), C12.52(C22×C6), C22.9(C22×C18), (C22×C18).30C22, C3.(C6×C4○D4), (C2×C4)⋊5(C2×C18), (C6×C4○D4).2C3, C6.47(C3×C4○D4), (C3×C4○D4).21C6, (C3×D4).20(C2×C6), (C2×C6).8(C22×C6), (C3×Q8).33(C2×C6), (C2×C12).156(C2×C6), (C22×C6).50(C2×C6), SmallGroup(288,370)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C18
G = < a,b,c,d | a18=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 282 in 246 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C22, C22 [×6], C22 [×6], C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C9, C12 [×8], C2×C6, C2×C6 [×6], C2×C6 [×6], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C18, C18 [×2], C18 [×6], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C2×C4○D4, C36 [×8], C2×C18, C2×C18 [×6], C2×C18 [×6], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C2×C36, C2×C36 [×15], D4×C9 [×12], Q8×C9 [×4], C22×C18 [×3], C6×C4○D4, C22×C36 [×3], D4×C18 [×3], Q8×C18, C9×C4○D4 [×8], C4○D4×C18
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C9, C2×C6 [×35], C4○D4 [×2], C24, C18 [×15], C22×C6 [×15], C2×C4○D4, C2×C18 [×35], C3×C4○D4 [×2], C23×C6, C22×C18 [×15], C6×C4○D4, C9×C4○D4 [×2], C23×C18, C4○D4×C18
►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)
(1 130 68 113)(2 131 69 114)(3 132 70 115)(4 133 71 116)(5 134 72 117)(6 135 55 118)(7 136 56 119)(8 137 57 120)(9 138 58 121)(10 139 59 122)(11 140 60 123)(12 141 61 124)(13 142 62 125)(14 143 63 126)(15 144 64 109)(16 127 65 110)(17 128 66 111)(18 129 67 112)(19 105 49 79)(20 106 50 80)(21 107 51 81)(22 108 52 82)(23 91 53 83)(24 92 54 84)(25 93 37 85)(26 94 38 86)(27 95 39 87)(28 96 40 88)(29 97 41 89)(30 98 42 90)(31 99 43 73)(32 100 44 74)(33 101 45 75)(34 102 46 76)(35 103 47 77)(36 104 48 78)
(1 139 68 122)(2 140 69 123)(3 141 70 124)(4 142 71 125)(5 143 72 126)(6 144 55 109)(7 127 56 110)(8 128 57 111)(9 129 58 112)(10 130 59 113)(11 131 60 114)(12 132 61 115)(13 133 62 116)(14 134 63 117)(15 135 64 118)(16 136 65 119)(17 137 66 120)(18 138 67 121)(19 88 49 96)(20 89 50 97)(21 90 51 98)(22 73 52 99)(23 74 53 100)(24 75 54 101)(25 76 37 102)(26 77 38 103)(27 78 39 104)(28 79 40 105)(29 80 41 106)(30 81 42 107)(31 82 43 108)(32 83 44 91)(33 84 45 92)(34 85 46 93)(35 86 47 94)(36 87 48 95)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 109)(33 110)(34 111)(35 112)(36 113)(37 137)(38 138)(39 139)(40 140)(41 141)(42 142)(43 143)(44 144)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(71 81)(72 82)
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), (1,130,68,113)(2,131,69,114)(3,132,70,115)(4,133,71,116)(5,134,72,117)(6,135,55,118)(7,136,56,119)(8,137,57,120)(9,138,58,121)(10,139,59,122)(11,140,60,123)(12,141,61,124)(13,142,62,125)(14,143,63,126)(15,144,64,109)(16,127,65,110)(17,128,66,111)(18,129,67,112)(19,105,49,79)(20,106,50,80)(21,107,51,81)(22,108,52,82)(23,91,53,83)(24,92,54,84)(25,93,37,85)(26,94,38,86)(27,95,39,87)(28,96,40,88)(29,97,41,89)(30,98,42,90)(31,99,43,73)(32,100,44,74)(33,101,45,75)(34,102,46,76)(35,103,47,77)(36,104,48,78), (1,139,68,122)(2,140,69,123)(3,141,70,124)(4,142,71,125)(5,143,72,126)(6,144,55,109)(7,127,56,110)(8,128,57,111)(9,129,58,112)(10,130,59,113)(11,131,60,114)(12,132,61,115)(13,133,62,116)(14,134,63,117)(15,135,64,118)(16,136,65,119)(17,137,66,120)(18,138,67,121)(19,88,49,96)(20,89,50,97)(21,90,51,98)(22,73,52,99)(23,74,53,100)(24,75,54,101)(25,76,37,102)(26,77,38,103)(27,78,39,104)(28,79,40,105)(29,80,41,106)(30,81,42,107)(31,82,43,108)(32,83,44,91)(33,84,45,92)(34,85,46,93)(35,86,47,94)(36,87,48,95), (1,104)(2,105)(3,106)(4,107)(5,108)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,109)(33,110)(34,111)(35,112)(36,113)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82)>;
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), (1,130,68,113)(2,131,69,114)(3,132,70,115)(4,133,71,116)(5,134,72,117)(6,135,55,118)(7,136,56,119)(8,137,57,120)(9,138,58,121)(10,139,59,122)(11,140,60,123)(12,141,61,124)(13,142,62,125)(14,143,63,126)(15,144,64,109)(16,127,65,110)(17,128,66,111)(18,129,67,112)(19,105,49,79)(20,106,50,80)(21,107,51,81)(22,108,52,82)(23,91,53,83)(24,92,54,84)(25,93,37,85)(26,94,38,86)(27,95,39,87)(28,96,40,88)(29,97,41,89)(30,98,42,90)(31,99,43,73)(32,100,44,74)(33,101,45,75)(34,102,46,76)(35,103,47,77)(36,104,48,78), (1,139,68,122)(2,140,69,123)(3,141,70,124)(4,142,71,125)(5,143,72,126)(6,144,55,109)(7,127,56,110)(8,128,57,111)(9,129,58,112)(10,130,59,113)(11,131,60,114)(12,132,61,115)(13,133,62,116)(14,134,63,117)(15,135,64,118)(16,136,65,119)(17,137,66,120)(18,138,67,121)(19,88,49,96)(20,89,50,97)(21,90,51,98)(22,73,52,99)(23,74,53,100)(24,75,54,101)(25,76,37,102)(26,77,38,103)(27,78,39,104)(28,79,40,105)(29,80,41,106)(30,81,42,107)(31,82,43,108)(32,83,44,91)(33,84,45,92)(34,85,46,93)(35,86,47,94)(36,87,48,95), (1,104)(2,105)(3,106)(4,107)(5,108)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,109)(33,110)(34,111)(35,112)(36,113)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82) );
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)], [(1,130,68,113),(2,131,69,114),(3,132,70,115),(4,133,71,116),(5,134,72,117),(6,135,55,118),(7,136,56,119),(8,137,57,120),(9,138,58,121),(10,139,59,122),(11,140,60,123),(12,141,61,124),(13,142,62,125),(14,143,63,126),(15,144,64,109),(16,127,65,110),(17,128,66,111),(18,129,67,112),(19,105,49,79),(20,106,50,80),(21,107,51,81),(22,108,52,82),(23,91,53,83),(24,92,54,84),(25,93,37,85),(26,94,38,86),(27,95,39,87),(28,96,40,88),(29,97,41,89),(30,98,42,90),(31,99,43,73),(32,100,44,74),(33,101,45,75),(34,102,46,76),(35,103,47,77),(36,104,48,78)], [(1,139,68,122),(2,140,69,123),(3,141,70,124),(4,142,71,125),(5,143,72,126),(6,144,55,109),(7,127,56,110),(8,128,57,111),(9,129,58,112),(10,130,59,113),(11,131,60,114),(12,132,61,115),(13,133,62,116),(14,134,63,117),(15,135,64,118),(16,136,65,119),(17,137,66,120),(18,138,67,121),(19,88,49,96),(20,89,50,97),(21,90,51,98),(22,73,52,99),(23,74,53,100),(24,75,54,101),(25,76,37,102),(26,77,38,103),(27,78,39,104),(28,79,40,105),(29,80,41,106),(30,81,42,107),(31,82,43,108),(32,83,44,91),(33,84,45,92),(34,85,46,93),(35,86,47,94),(36,87,48,95)], [(1,104),(2,105),(3,106),(4,107),(5,108),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,99),(15,100),(16,101),(17,102),(18,103),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,124),(30,125),(31,126),(32,109),(33,110),(34,111),(35,112),(36,113),(37,137),(38,138),(39,139),(40,140),(41,141),(42,142),(43,143),(44,144),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,83),(56,84),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(71,81),(72,82)])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6F | 6G | ··· | 6R | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12T | 18A | ··· | 18R | 18S | ··· | 18BB | 36A | ··· | 36X | 36Y | ··· | 36BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | C4○D4 | C3×C4○D4 | C9×C4○D4 |
| kernel | C4○D4×C18 | C22×C36 | D4×C18 | Q8×C18 | C9×C4○D4 | C6×C4○D4 | C22×C12 | C6×D4 | C6×Q8 | C3×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C18 | C6 | C2 |
| # reps | 1 | 3 | 3 | 1 | 8 | 2 | 6 | 6 | 2 | 16 | 6 | 18 | 18 | 6 | 48 | 4 | 8 | 24 |
Matrix representation of C4○D4×C18 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 21 | 0 |
| 0 | 0 | 21 |
| 1 | 0 | 0 |
| 0 | 31 | 0 |
| 0 | 0 | 31 |
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 13 | 31 |
| 1 | 0 | 0 |
| 0 | 3 | 20 |
| 0 | 7 | 34 |
G:=sub<GL(3,GF(37))| [36,0,0,0,21,0,0,0,21],[1,0,0,0,31,0,0,0,31],[1,0,0,0,6,13,0,0,31],[1,0,0,0,3,7,0,20,34] >;
C4○D4×C18 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{18} % in TeX
G:=Group("C4oD4xC18"); // GroupNames label
G:=SmallGroup(288,370);
// by ID
G=gap.SmallGroup(288,370);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,268,242]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations