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, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C9, C12, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C18, C18, C18, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4○D4, C36, C2×C18, C2×C18, C2×C18, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2×C36, C2×C36, D4×C9, Q8×C9, C22×C18, C6×C4○D4, C22×C36, D4×C18, Q8×C18, C9×C4○D4, C4○D4×C18
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C4○D4, C24, C18, C22×C6, C2×C4○D4, C2×C18, C3×C4○D4, C23×C6, C22×C18, C6×C4○D4, C9×C4○D4, 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 60 132 123)(2 61 133 124)(3 62 134 125)(4 63 135 126)(5 64 136 109)(6 65 137 110)(7 66 138 111)(8 67 139 112)(9 68 140 113)(10 69 141 114)(11 70 142 115)(12 71 143 116)(13 72 144 117)(14 55 127 118)(15 56 128 119)(16 57 129 120)(17 58 130 121)(18 59 131 122)(19 74 50 107)(20 75 51 108)(21 76 52 91)(22 77 53 92)(23 78 54 93)(24 79 37 94)(25 80 38 95)(26 81 39 96)(27 82 40 97)(28 83 41 98)(29 84 42 99)(30 85 43 100)(31 86 44 101)(32 87 45 102)(33 88 46 103)(34 89 47 104)(35 90 48 105)(36 73 49 106)
(1 69 132 114)(2 70 133 115)(3 71 134 116)(4 72 135 117)(5 55 136 118)(6 56 137 119)(7 57 138 120)(8 58 139 121)(9 59 140 122)(10 60 141 123)(11 61 142 124)(12 62 143 125)(13 63 144 126)(14 64 127 109)(15 65 128 110)(16 66 129 111)(17 67 130 112)(18 68 131 113)(19 98 50 83)(20 99 51 84)(21 100 52 85)(22 101 53 86)(23 102 54 87)(24 103 37 88)(25 104 38 89)(26 105 39 90)(27 106 40 73)(28 107 41 74)(29 108 42 75)(30 91 43 76)(31 92 44 77)(32 93 45 78)(33 94 46 79)(34 95 47 80)(35 96 48 81)(36 97 49 82)
(1 33)(2 34)(3 35)(4 36)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(37 141)(38 142)(39 143)(40 144)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(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)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 109)(108 110)
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), (1,60,132,123)(2,61,133,124)(3,62,134,125)(4,63,135,126)(5,64,136,109)(6,65,137,110)(7,66,138,111)(8,67,139,112)(9,68,140,113)(10,69,141,114)(11,70,142,115)(12,71,143,116)(13,72,144,117)(14,55,127,118)(15,56,128,119)(16,57,129,120)(17,58,130,121)(18,59,131,122)(19,74,50,107)(20,75,51,108)(21,76,52,91)(22,77,53,92)(23,78,54,93)(24,79,37,94)(25,80,38,95)(26,81,39,96)(27,82,40,97)(28,83,41,98)(29,84,42,99)(30,85,43,100)(31,86,44,101)(32,87,45,102)(33,88,46,103)(34,89,47,104)(35,90,48,105)(36,73,49,106), (1,69,132,114)(2,70,133,115)(3,71,134,116)(4,72,135,117)(5,55,136,118)(6,56,137,119)(7,57,138,120)(8,58,139,121)(9,59,140,122)(10,60,141,123)(11,61,142,124)(12,62,143,125)(13,63,144,126)(14,64,127,109)(15,65,128,110)(16,66,129,111)(17,67,130,112)(18,68,131,113)(19,98,50,83)(20,99,51,84)(21,100,52,85)(22,101,53,86)(23,102,54,87)(24,103,37,88)(25,104,38,89)(26,105,39,90)(27,106,40,73)(28,107,41,74)(29,108,42,75)(30,91,43,76)(31,92,44,77)(32,93,45,78)(33,94,46,79)(34,95,47,80)(35,96,48,81)(36,97,49,82), (1,33)(2,34)(3,35)(4,36)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(37,141)(38,142)(39,143)(40,144)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(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)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,109)(108,110)>;
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), (1,60,132,123)(2,61,133,124)(3,62,134,125)(4,63,135,126)(5,64,136,109)(6,65,137,110)(7,66,138,111)(8,67,139,112)(9,68,140,113)(10,69,141,114)(11,70,142,115)(12,71,143,116)(13,72,144,117)(14,55,127,118)(15,56,128,119)(16,57,129,120)(17,58,130,121)(18,59,131,122)(19,74,50,107)(20,75,51,108)(21,76,52,91)(22,77,53,92)(23,78,54,93)(24,79,37,94)(25,80,38,95)(26,81,39,96)(27,82,40,97)(28,83,41,98)(29,84,42,99)(30,85,43,100)(31,86,44,101)(32,87,45,102)(33,88,46,103)(34,89,47,104)(35,90,48,105)(36,73,49,106), (1,69,132,114)(2,70,133,115)(3,71,134,116)(4,72,135,117)(5,55,136,118)(6,56,137,119)(7,57,138,120)(8,58,139,121)(9,59,140,122)(10,60,141,123)(11,61,142,124)(12,62,143,125)(13,63,144,126)(14,64,127,109)(15,65,128,110)(16,66,129,111)(17,67,130,112)(18,68,131,113)(19,98,50,83)(20,99,51,84)(21,100,52,85)(22,101,53,86)(23,102,54,87)(24,103,37,88)(25,104,38,89)(26,105,39,90)(27,106,40,73)(28,107,41,74)(29,108,42,75)(30,91,43,76)(31,92,44,77)(32,93,45,78)(33,94,46,79)(34,95,47,80)(35,96,48,81)(36,97,49,82), (1,33)(2,34)(3,35)(4,36)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(37,141)(38,142)(39,143)(40,144)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(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)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,109)(108,110) );
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)], [(1,60,132,123),(2,61,133,124),(3,62,134,125),(4,63,135,126),(5,64,136,109),(6,65,137,110),(7,66,138,111),(8,67,139,112),(9,68,140,113),(10,69,141,114),(11,70,142,115),(12,71,143,116),(13,72,144,117),(14,55,127,118),(15,56,128,119),(16,57,129,120),(17,58,130,121),(18,59,131,122),(19,74,50,107),(20,75,51,108),(21,76,52,91),(22,77,53,92),(23,78,54,93),(24,79,37,94),(25,80,38,95),(26,81,39,96),(27,82,40,97),(28,83,41,98),(29,84,42,99),(30,85,43,100),(31,86,44,101),(32,87,45,102),(33,88,46,103),(34,89,47,104),(35,90,48,105),(36,73,49,106)], [(1,69,132,114),(2,70,133,115),(3,71,134,116),(4,72,135,117),(5,55,136,118),(6,56,137,119),(7,57,138,120),(8,58,139,121),(9,59,140,122),(10,60,141,123),(11,61,142,124),(12,62,143,125),(13,63,144,126),(14,64,127,109),(15,65,128,110),(16,66,129,111),(17,67,130,112),(18,68,131,113),(19,98,50,83),(20,99,51,84),(21,100,52,85),(22,101,53,86),(23,102,54,87),(24,103,37,88),(25,104,38,89),(26,105,39,90),(27,106,40,73),(28,107,41,74),(29,108,42,75),(30,91,43,76),(31,92,44,77),(32,93,45,78),(33,94,46,79),(34,95,47,80),(35,96,48,81),(36,97,49,82)], [(1,33),(2,34),(3,35),(4,36),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(37,141),(38,142),(39,143),(40,144),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(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),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,109),(108,110)]])
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