direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×D4⋊C4, D4⋊1C36, C18.13D8, C36.60D4, C18.9SD16, C4⋊C4⋊1C18, (C2×C72)⋊4C2, (C2×C8)⋊2C18, (D4×C9)⋊4C4, C2.1(C9×D8), (C2×C24).2C6, C4.1(C2×C36), (C6×D4).8C6, C4.11(D4×C9), C6.13(C3×D8), C36.30(C2×C4), (C2×D4).3C18, (C3×D4).2C12, (D4×C18).9C2, C12.69(C3×D4), (C2×C18).46D4, C2.1(C9×SD16), C6.9(C3×SD16), C22.8(D4×C9), C12.30(C2×C12), C18.24(C22⋊C4), (C2×C36).113C22, (C9×C4⋊C4)⋊10C2, (C3×C4⋊C4).8C6, C3.(C3×D4⋊C4), (C3×D4⋊C4).C3, (C2×C6).55(C3×D4), C2.6(C9×C22⋊C4), (C2×C4).12(C2×C18), C6.24(C3×C22⋊C4), (C2×C12).133(C2×C6), SmallGroup(288,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×D4⋊C4
G = < a,b,c,d | a9=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Subgroups: 138 in 75 conjugacy classes, 42 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, C9, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C18, C18, C24, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C36, C36, C2×C18, C2×C18, C3×C4⋊C4, C2×C24, C6×D4, C72, C2×C36, C2×C36, D4×C9, D4×C9, C22×C18, C3×D4⋊C4, C9×C4⋊C4, C2×C72, D4×C18, C9×D4⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, D8, SD16, C18, C2×C12, C3×D4, D4⋊C4, C36, C2×C18, C3×C22⋊C4, C3×D8, C3×SD16, C2×C36, D4×C9, C3×D4⋊C4, C9×C22⋊C4, C9×D8, C9×SD16, C9×D4⋊C4
►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 55 75 107)(2 56 76 108)(3 57 77 100)(4 58 78 101)(5 59 79 102)(6 60 80 103)(7 61 81 104)(8 62 73 105)(9 63 74 106)(10 84 42 67)(11 85 43 68)(12 86 44 69)(13 87 45 70)(14 88 37 71)(15 89 38 72)(16 90 39 64)(17 82 40 65)(18 83 41 66)(19 33 109 94)(20 34 110 95)(21 35 111 96)(22 36 112 97)(23 28 113 98)(24 29 114 99)(25 30 115 91)(26 31 116 92)(27 32 117 93)(46 132 142 121)(47 133 143 122)(48 134 144 123)(49 135 136 124)(50 127 137 125)(51 128 138 126)(52 129 139 118)(53 130 140 119)(54 131 141 120)
(1 107)(2 108)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 64)(17 65)(18 66)(28 98)(29 99)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 88)(38 89)(39 90)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 73)(63 74)(118 129)(119 130)(120 131)(121 132)(122 133)(123 134)(124 135)(125 127)(126 128)
(1 129 10 98)(2 130 11 99)(3 131 12 91)(4 132 13 92)(5 133 14 93)(6 134 15 94)(7 135 16 95)(8 127 17 96)(9 128 18 97)(19 103 144 72)(20 104 136 64)(21 105 137 65)(22 106 138 66)(23 107 139 67)(24 108 140 68)(25 100 141 69)(26 101 142 70)(27 102 143 71)(28 75 118 42)(29 76 119 43)(30 77 120 44)(31 78 121 45)(32 79 122 37)(33 80 123 38)(34 81 124 39)(35 73 125 40)(36 74 126 41)(46 87 116 58)(47 88 117 59)(48 89 109 60)(49 90 110 61)(50 82 111 62)(51 83 112 63)(52 84 113 55)(53 85 114 56)(54 86 115 57)
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), (1,55,75,107)(2,56,76,108)(3,57,77,100)(4,58,78,101)(5,59,79,102)(6,60,80,103)(7,61,81,104)(8,62,73,105)(9,63,74,106)(10,84,42,67)(11,85,43,68)(12,86,44,69)(13,87,45,70)(14,88,37,71)(15,89,38,72)(16,90,39,64)(17,82,40,65)(18,83,41,66)(19,33,109,94)(20,34,110,95)(21,35,111,96)(22,36,112,97)(23,28,113,98)(24,29,114,99)(25,30,115,91)(26,31,116,92)(27,32,117,93)(46,132,142,121)(47,133,143,122)(48,134,144,123)(49,135,136,124)(50,127,137,125)(51,128,138,126)(52,129,139,118)(53,130,140,119)(54,131,141,120), (1,107)(2,108)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,64)(17,65)(18,66)(28,98)(29,99)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,88)(38,89)(39,90)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,73)(63,74)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,127)(126,128), (1,129,10,98)(2,130,11,99)(3,131,12,91)(4,132,13,92)(5,133,14,93)(6,134,15,94)(7,135,16,95)(8,127,17,96)(9,128,18,97)(19,103,144,72)(20,104,136,64)(21,105,137,65)(22,106,138,66)(23,107,139,67)(24,108,140,68)(25,100,141,69)(26,101,142,70)(27,102,143,71)(28,75,118,42)(29,76,119,43)(30,77,120,44)(31,78,121,45)(32,79,122,37)(33,80,123,38)(34,81,124,39)(35,73,125,40)(36,74,126,41)(46,87,116,58)(47,88,117,59)(48,89,109,60)(49,90,110,61)(50,82,111,62)(51,83,112,63)(52,84,113,55)(53,85,114,56)(54,86,115,57)>;
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), (1,55,75,107)(2,56,76,108)(3,57,77,100)(4,58,78,101)(5,59,79,102)(6,60,80,103)(7,61,81,104)(8,62,73,105)(9,63,74,106)(10,84,42,67)(11,85,43,68)(12,86,44,69)(13,87,45,70)(14,88,37,71)(15,89,38,72)(16,90,39,64)(17,82,40,65)(18,83,41,66)(19,33,109,94)(20,34,110,95)(21,35,111,96)(22,36,112,97)(23,28,113,98)(24,29,114,99)(25,30,115,91)(26,31,116,92)(27,32,117,93)(46,132,142,121)(47,133,143,122)(48,134,144,123)(49,135,136,124)(50,127,137,125)(51,128,138,126)(52,129,139,118)(53,130,140,119)(54,131,141,120), (1,107)(2,108)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,64)(17,65)(18,66)(28,98)(29,99)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,88)(38,89)(39,90)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,73)(63,74)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,127)(126,128), (1,129,10,98)(2,130,11,99)(3,131,12,91)(4,132,13,92)(5,133,14,93)(6,134,15,94)(7,135,16,95)(8,127,17,96)(9,128,18,97)(19,103,144,72)(20,104,136,64)(21,105,137,65)(22,106,138,66)(23,107,139,67)(24,108,140,68)(25,100,141,69)(26,101,142,70)(27,102,143,71)(28,75,118,42)(29,76,119,43)(30,77,120,44)(31,78,121,45)(32,79,122,37)(33,80,123,38)(34,81,124,39)(35,73,125,40)(36,74,126,41)(46,87,116,58)(47,88,117,59)(48,89,109,60)(49,90,110,61)(50,82,111,62)(51,83,112,63)(52,84,113,55)(53,85,114,56)(54,86,115,57) );
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)], [(1,55,75,107),(2,56,76,108),(3,57,77,100),(4,58,78,101),(5,59,79,102),(6,60,80,103),(7,61,81,104),(8,62,73,105),(9,63,74,106),(10,84,42,67),(11,85,43,68),(12,86,44,69),(13,87,45,70),(14,88,37,71),(15,89,38,72),(16,90,39,64),(17,82,40,65),(18,83,41,66),(19,33,109,94),(20,34,110,95),(21,35,111,96),(22,36,112,97),(23,28,113,98),(24,29,114,99),(25,30,115,91),(26,31,116,92),(27,32,117,93),(46,132,142,121),(47,133,143,122),(48,134,144,123),(49,135,136,124),(50,127,137,125),(51,128,138,126),(52,129,139,118),(53,130,140,119),(54,131,141,120)], [(1,107),(2,108),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,64),(17,65),(18,66),(28,98),(29,99),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,88),(38,89),(39,90),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,73),(63,74),(118,129),(119,130),(120,131),(121,132),(122,133),(123,134),(124,135),(125,127),(126,128)], [(1,129,10,98),(2,130,11,99),(3,131,12,91),(4,132,13,92),(5,133,14,93),(6,134,15,94),(7,135,16,95),(8,127,17,96),(9,128,18,97),(19,103,144,72),(20,104,136,64),(21,105,137,65),(22,106,138,66),(23,107,139,67),(24,108,140,68),(25,100,141,69),(26,101,142,70),(27,102,143,71),(28,75,118,42),(29,76,119,43),(30,77,120,44),(31,78,121,45),(32,79,122,37),(33,80,123,38),(34,81,124,39),(35,73,125,40),(36,74,126,41),(46,87,116,58),(47,88,117,59),(48,89,109,60),(49,90,110,61),(50,82,111,62),(51,83,112,63),(52,84,113,55),(53,85,114,56),(54,86,115,57)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18R | 18S | ··· | 18AD | 24A | ··· | 24H | 36A | ··· | 36L | 36M | ··· | 36X | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C36 | D4 | D4 | D8 | SD16 | C3×D4 | C3×D4 | C3×D8 | C3×SD16 | D4×C9 | D4×C9 | C9×D8 | C9×SD16 |
| kernel | C9×D4⋊C4 | C9×C4⋊C4 | C2×C72 | D4×C18 | C3×D4⋊C4 | D4×C9 | C3×C4⋊C4 | C2×C24 | C6×D4 | D4⋊C4 | C3×D4 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C36 | C2×C18 | C18 | C18 | C12 | C2×C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 24 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 |
Matrix representation of C9×D4⋊C4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 |
| 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 72 |
| 60 | 48 | 0 | 0 |
| 36 | 13 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 6 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[72,0,0,0,0,72,0,0,0,0,1,1,0,0,71,72],[72,39,0,0,0,1,0,0,0,0,1,0,0,0,71,72],[60,36,0,0,48,13,0,0,0,0,0,6,0,0,12,0] >;
C9×D4⋊C4 in GAP, Magma, Sage, TeX
C_9\times D_4\rtimes C_4
% in TeX
G:=Group("C9xD4:C4"); // GroupNames label
G:=SmallGroup(288,52);
// by ID
G=gap.SmallGroup(288,52);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2194,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations