direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C4⋊1D4, C36⋊6D4, C42⋊9C18, C4⋊1(D4×C9), (C4×C36)⋊13C2, (C2×D4)⋊3C18, C6.72(C6×D4), C2.9(D4×C18), (D4×C18)⋊12C2, (C4×C12).20C6, (C6×D4).13C6, C18.72(C2×D4), C12.40(C3×D4), C23.6(C2×C18), (C2×C18).82C23, (C2×C36).124C22, (C22×C18).4C22, C22.17(C22×C18), C3.(C3×C4⋊1D4), (C2×C4).22(C2×C18), (C3×C4⋊1D4).2C3, (C22×C6).9(C2×C6), (C2×C12).140(C2×C6), (C2×C6).87(C22×C6), SmallGroup(288,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4⋊1D4
G = < a,b,c,d | a9=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 270 in 162 conjugacy classes, 78 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, D4, C23, C9, C12, C2×C6, C2×C6, C42, C2×D4, C18, C18, C2×C12, C3×D4, C22×C6, C4⋊1D4, C36, C2×C18, C2×C18, C4×C12, C6×D4, C2×C36, D4×C9, C22×C18, C3×C4⋊1D4, C4×C36, D4×C18, C9×C4⋊1D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C18, C3×D4, C22×C6, C4⋊1D4, C2×C18, C6×D4, D4×C9, C22×C18, C3×C4⋊1D4, D4×C18, C9×C4⋊1D4
►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)
(1 129 47 142)(2 130 48 143)(3 131 49 144)(4 132 50 136)(5 133 51 137)(6 134 52 138)(7 135 53 139)(8 127 54 140)(9 128 46 141)(10 30 22 40)(11 31 23 41)(12 32 24 42)(13 33 25 43)(14 34 26 44)(15 35 27 45)(16 36 19 37)(17 28 20 38)(18 29 21 39)(55 104 68 91)(56 105 69 92)(57 106 70 93)(58 107 71 94)(59 108 72 95)(60 100 64 96)(61 101 65 97)(62 102 66 98)(63 103 67 99)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)
(1 70 34 74)(2 71 35 75)(3 72 36 76)(4 64 28 77)(5 65 29 78)(6 66 30 79)(7 67 31 80)(8 68 32 81)(9 69 33 73)(10 115 138 102)(11 116 139 103)(12 117 140 104)(13 109 141 105)(14 110 142 106)(15 111 143 107)(16 112 144 108)(17 113 136 100)(18 114 137 101)(19 121 131 95)(20 122 132 96)(21 123 133 97)(22 124 134 98)(23 125 135 99)(24 126 127 91)(25 118 128 92)(26 119 129 93)(27 120 130 94)(37 85 49 59)(38 86 50 60)(39 87 51 61)(40 88 52 62)(41 89 53 63)(42 90 54 55)(43 82 46 56)(44 83 47 57)(45 84 48 58)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 46)(26 47)(27 48)(28 136)(29 137)(30 138)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 131)(38 132)(39 133)(40 134)(41 135)(42 127)(43 128)(44 129)(45 130)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)
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), (1,129,47,142)(2,130,48,143)(3,131,49,144)(4,132,50,136)(5,133,51,137)(6,134,52,138)(7,135,53,139)(8,127,54,140)(9,128,46,141)(10,30,22,40)(11,31,23,41)(12,32,24,42)(13,33,25,43)(14,34,26,44)(15,35,27,45)(16,36,19,37)(17,28,20,38)(18,29,21,39)(55,104,68,91)(56,105,69,92)(57,106,70,93)(58,107,71,94)(59,108,72,95)(60,100,64,96)(61,101,65,97)(62,102,66,98)(63,103,67,99)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117), (1,70,34,74)(2,71,35,75)(3,72,36,76)(4,64,28,77)(5,65,29,78)(6,66,30,79)(7,67,31,80)(8,68,32,81)(9,69,33,73)(10,115,138,102)(11,116,139,103)(12,117,140,104)(13,109,141,105)(14,110,142,106)(15,111,143,107)(16,112,144,108)(17,113,136,100)(18,114,137,101)(19,121,131,95)(20,122,132,96)(21,123,133,97)(22,124,134,98)(23,125,135,99)(24,126,127,91)(25,118,128,92)(26,119,129,93)(27,120,130,94)(37,85,49,59)(38,86,50,60)(39,87,51,61)(40,88,52,62)(41,89,53,63)(42,90,54,55)(43,82,46,56)(44,83,47,57)(45,84,48,58), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,46)(26,47)(27,48)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,131)(38,132)(39,133)(40,134)(41,135)(42,127)(43,128)(44,129)(45,130)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)>;
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), (1,129,47,142)(2,130,48,143)(3,131,49,144)(4,132,50,136)(5,133,51,137)(6,134,52,138)(7,135,53,139)(8,127,54,140)(9,128,46,141)(10,30,22,40)(11,31,23,41)(12,32,24,42)(13,33,25,43)(14,34,26,44)(15,35,27,45)(16,36,19,37)(17,28,20,38)(18,29,21,39)(55,104,68,91)(56,105,69,92)(57,106,70,93)(58,107,71,94)(59,108,72,95)(60,100,64,96)(61,101,65,97)(62,102,66,98)(63,103,67,99)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117), (1,70,34,74)(2,71,35,75)(3,72,36,76)(4,64,28,77)(5,65,29,78)(6,66,30,79)(7,67,31,80)(8,68,32,81)(9,69,33,73)(10,115,138,102)(11,116,139,103)(12,117,140,104)(13,109,141,105)(14,110,142,106)(15,111,143,107)(16,112,144,108)(17,113,136,100)(18,114,137,101)(19,121,131,95)(20,122,132,96)(21,123,133,97)(22,124,134,98)(23,125,135,99)(24,126,127,91)(25,118,128,92)(26,119,129,93)(27,120,130,94)(37,85,49,59)(38,86,50,60)(39,87,51,61)(40,88,52,62)(41,89,53,63)(42,90,54,55)(43,82,46,56)(44,83,47,57)(45,84,48,58), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,46)(26,47)(27,48)(28,136)(29,137)(30,138)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,131)(38,132)(39,133)(40,134)(41,135)(42,127)(43,128)(44,129)(45,130)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126) );
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)], [(1,129,47,142),(2,130,48,143),(3,131,49,144),(4,132,50,136),(5,133,51,137),(6,134,52,138),(7,135,53,139),(8,127,54,140),(9,128,46,141),(10,30,22,40),(11,31,23,41),(12,32,24,42),(13,33,25,43),(14,34,26,44),(15,35,27,45),(16,36,19,37),(17,28,20,38),(18,29,21,39),(55,104,68,91),(56,105,69,92),(57,106,70,93),(58,107,71,94),(59,108,72,95),(60,100,64,96),(61,101,65,97),(62,102,66,98),(63,103,67,99),(73,118,82,109),(74,119,83,110),(75,120,84,111),(76,121,85,112),(77,122,86,113),(78,123,87,114),(79,124,88,115),(80,125,89,116),(81,126,90,117)], [(1,70,34,74),(2,71,35,75),(3,72,36,76),(4,64,28,77),(5,65,29,78),(6,66,30,79),(7,67,31,80),(8,68,32,81),(9,69,33,73),(10,115,138,102),(11,116,139,103),(12,117,140,104),(13,109,141,105),(14,110,142,106),(15,111,143,107),(16,112,144,108),(17,113,136,100),(18,114,137,101),(19,121,131,95),(20,122,132,96),(21,123,133,97),(22,124,134,98),(23,125,135,99),(24,126,127,91),(25,118,128,92),(26,119,129,93),(27,120,130,94),(37,85,49,59),(38,86,50,60),(39,87,51,61),(40,88,52,62),(41,89,53,63),(42,90,54,55),(43,82,46,56),(44,83,47,57),(45,84,48,58)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,46),(26,47),(27,48),(28,136),(29,137),(30,138),(31,139),(32,140),(33,141),(34,142),(35,143),(36,144),(37,131),(38,132),(39,133),(40,134),(41,135),(42,127),(43,128),(44,129),(45,130),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,121),(86,122),(87,123),(88,124),(89,125),(90,126)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 6A | ··· | 6F | 6G | ··· | 6N | 9A | ··· | 9F | 12A | ··· | 12L | 18A | ··· | 18R | 18S | ··· | 18AP | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D4 | C3×D4 | D4×C9 |
| kernel | C9×C4⋊1D4 | C4×C36 | D4×C18 | C3×C4⋊1D4 | C4×C12 | C6×D4 | C4⋊1D4 | C42 | C2×D4 | C36 | C12 | C4 |
| # reps | 1 | 1 | 6 | 2 | 2 | 12 | 6 | 6 | 36 | 6 | 12 | 36 |
Matrix representation of C9×C4⋊1D4 ►in GL4(𝔽37) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [7,0,0,0,0,7,0,0,0,0,9,0,0,0,0,9],[0,1,0,0,36,0,0,0,0,0,36,0,0,0,0,36],[0,36,0,0,1,0,0,0,0,0,0,36,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,36] >;
C9×C4⋊1D4 in GAP, Magma, Sage, TeX
C_9\times C_4\rtimes_1D_4
% in TeX
G:=Group("C9xC4:1D4"); // GroupNames label
G:=SmallGroup(288,177);
// by ID
G=gap.SmallGroup(288,177);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,176,1094,268,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations