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 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×2], D4, C23, C9, C12 [×2], C12, C2×C6, C2×C6 [×4], C4⋊C4, C2×C8, C2×D4, C18 [×3], C18 [×2], C24, C2×C12, C2×C12, C3×D4 [×2], C3×D4, C22×C6, D4⋊C4, C36 [×2], C36, C2×C18, C2×C18 [×4], C3×C4⋊C4, C2×C24, C6×D4, C72, C2×C36, C2×C36, D4×C9 [×2], D4×C9, C22×C18, C3×D4⋊C4, C9×C4⋊C4, C2×C72, D4×C18, C9×D4⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, D8, SD16, C18 [×3], C2×C12, C3×D4 [×2], D4⋊C4, C36 [×2], C2×C18, C3×C22⋊C4, C3×D8, C3×SD16, C2×C36, D4×C9 [×2], 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 12 77 107)(2 13 78 108)(3 14 79 100)(4 15 80 101)(5 16 81 102)(6 17 73 103)(7 18 74 104)(8 10 75 105)(9 11 76 106)(19 97 115 29)(20 98 116 30)(21 99 117 31)(22 91 109 32)(23 92 110 33)(24 93 111 34)(25 94 112 35)(26 95 113 36)(27 96 114 28)(37 71 57 90)(38 72 58 82)(39 64 59 83)(40 65 60 84)(41 66 61 85)(42 67 62 86)(43 68 63 87)(44 69 55 88)(45 70 56 89)(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 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 73)(18 74)(19 115)(20 116)(21 117)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(37 90)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(55 69)(56 70)(57 71)(58 72)(59 64)(60 65)(61 66)(62 67)(63 68)(118 129)(119 130)(120 131)(121 132)(122 133)(123 134)(124 135)(125 127)(126 128)
(1 129 62 111)(2 130 63 112)(3 131 55 113)(4 132 56 114)(5 133 57 115)(6 134 58 116)(7 135 59 117)(8 127 60 109)(9 128 61 110)(10 50 84 91)(11 51 85 92)(12 52 86 93)(13 53 87 94)(14 54 88 95)(15 46 89 96)(16 47 90 97)(17 48 82 98)(18 49 83 99)(19 81 122 37)(20 73 123 38)(21 74 124 39)(22 75 125 40)(23 76 126 41)(24 77 118 42)(25 78 119 43)(26 79 120 44)(27 80 121 45)(28 101 142 70)(29 102 143 71)(30 103 144 72)(31 104 136 64)(32 105 137 65)(33 106 138 66)(34 107 139 67)(35 108 140 68)(36 100 141 69)
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), (1,12,77,107)(2,13,78,108)(3,14,79,100)(4,15,80,101)(5,16,81,102)(6,17,73,103)(7,18,74,104)(8,10,75,105)(9,11,76,106)(19,97,115,29)(20,98,116,30)(21,99,117,31)(22,91,109,32)(23,92,110,33)(24,93,111,34)(25,94,112,35)(26,95,113,36)(27,96,114,28)(37,71,57,90)(38,72,58,82)(39,64,59,83)(40,65,60,84)(41,66,61,85)(42,67,62,86)(43,68,63,87)(44,69,55,88)(45,70,56,89)(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)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,73)(18,74)(19,115)(20,116)(21,117)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(37,90)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(55,69)(56,70)(57,71)(58,72)(59,64)(60,65)(61,66)(62,67)(63,68)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,127)(126,128), (1,129,62,111)(2,130,63,112)(3,131,55,113)(4,132,56,114)(5,133,57,115)(6,134,58,116)(7,135,59,117)(8,127,60,109)(9,128,61,110)(10,50,84,91)(11,51,85,92)(12,52,86,93)(13,53,87,94)(14,54,88,95)(15,46,89,96)(16,47,90,97)(17,48,82,98)(18,49,83,99)(19,81,122,37)(20,73,123,38)(21,74,124,39)(22,75,125,40)(23,76,126,41)(24,77,118,42)(25,78,119,43)(26,79,120,44)(27,80,121,45)(28,101,142,70)(29,102,143,71)(30,103,144,72)(31,104,136,64)(32,105,137,65)(33,106,138,66)(34,107,139,67)(35,108,140,68)(36,100,141,69)>;
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), (1,12,77,107)(2,13,78,108)(3,14,79,100)(4,15,80,101)(5,16,81,102)(6,17,73,103)(7,18,74,104)(8,10,75,105)(9,11,76,106)(19,97,115,29)(20,98,116,30)(21,99,117,31)(22,91,109,32)(23,92,110,33)(24,93,111,34)(25,94,112,35)(26,95,113,36)(27,96,114,28)(37,71,57,90)(38,72,58,82)(39,64,59,83)(40,65,60,84)(41,66,61,85)(42,67,62,86)(43,68,63,87)(44,69,55,88)(45,70,56,89)(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)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,81)(17,73)(18,74)(19,115)(20,116)(21,117)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(37,90)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(55,69)(56,70)(57,71)(58,72)(59,64)(60,65)(61,66)(62,67)(63,68)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,127)(126,128), (1,129,62,111)(2,130,63,112)(3,131,55,113)(4,132,56,114)(5,133,57,115)(6,134,58,116)(7,135,59,117)(8,127,60,109)(9,128,61,110)(10,50,84,91)(11,51,85,92)(12,52,86,93)(13,53,87,94)(14,54,88,95)(15,46,89,96)(16,47,90,97)(17,48,82,98)(18,49,83,99)(19,81,122,37)(20,73,123,38)(21,74,124,39)(22,75,125,40)(23,76,126,41)(24,77,118,42)(25,78,119,43)(26,79,120,44)(27,80,121,45)(28,101,142,70)(29,102,143,71)(30,103,144,72)(31,104,136,64)(32,105,137,65)(33,106,138,66)(34,107,139,67)(35,108,140,68)(36,100,141,69) );
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)], [(1,12,77,107),(2,13,78,108),(3,14,79,100),(4,15,80,101),(5,16,81,102),(6,17,73,103),(7,18,74,104),(8,10,75,105),(9,11,76,106),(19,97,115,29),(20,98,116,30),(21,99,117,31),(22,91,109,32),(23,92,110,33),(24,93,111,34),(25,94,112,35),(26,95,113,36),(27,96,114,28),(37,71,57,90),(38,72,58,82),(39,64,59,83),(40,65,60,84),(41,66,61,85),(42,67,62,86),(43,68,63,87),(44,69,55,88),(45,70,56,89),(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),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,81),(17,73),(18,74),(19,115),(20,116),(21,117),(22,109),(23,110),(24,111),(25,112),(26,113),(27,114),(37,90),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88),(45,89),(55,69),(56,70),(57,71),(58,72),(59,64),(60,65),(61,66),(62,67),(63,68),(118,129),(119,130),(120,131),(121,132),(122,133),(123,134),(124,135),(125,127),(126,128)], [(1,129,62,111),(2,130,63,112),(3,131,55,113),(4,132,56,114),(5,133,57,115),(6,134,58,116),(7,135,59,117),(8,127,60,109),(9,128,61,110),(10,50,84,91),(11,51,85,92),(12,52,86,93),(13,53,87,94),(14,54,88,95),(15,46,89,96),(16,47,90,97),(17,48,82,98),(18,49,83,99),(19,81,122,37),(20,73,123,38),(21,74,124,39),(22,75,125,40),(23,76,126,41),(24,77,118,42),(25,78,119,43),(26,79,120,44),(27,80,121,45),(28,101,142,70),(29,102,143,71),(30,103,144,72),(31,104,136,64),(32,105,137,65),(33,106,138,66),(34,107,139,67),(35,108,140,68),(36,100,141,69)])
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