direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C4.4D4, C42⋊8C18, C36.40D4, C4.4(D4×C9), (C4×C36)⋊12C2, (C2×Q8)⋊4C18, (Q8×C18)⋊9C2, C2.8(D4×C18), C6.71(C6×D4), C22⋊C4⋊5C18, (C4×C12).18C6, (C2×D4).5C18, (C6×D4).12C6, C18.71(C2×D4), C12.39(C3×D4), (C6×Q8).17C6, (D4×C18).12C2, C23.4(C2×C18), C18.44(C4○D4), (C2×C18).79C23, (C2×C36).80C22, (C22×C18).2C22, C22.14(C22×C18), C2.7(C9×C4○D4), C3.(C3×C4.4D4), C6.44(C3×C4○D4), (C3×C4.4D4).C3, (C9×C22⋊C4)⋊13C2, (C2×C12).83(C2×C6), (C2×C4).20(C2×C18), (C3×C22⋊C4).9C6, (C22×C6).7(C2×C6), (C2×C6).84(C22×C6), SmallGroup(288,174)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4.4D4
G = < a,b,c,d | a9=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 174 in 114 conjugacy classes, 66 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C9, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C18, C18, C18, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.4D4, C36, C36, C2×C18, C2×C18, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C2×C36, C2×C36, D4×C9, Q8×C9, C22×C18, C3×C4.4D4, C4×C36, C9×C22⋊C4, D4×C18, Q8×C18, C9×C4.4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C4○D4, C18, C3×D4, C22×C6, C4.4D4, C2×C18, C6×D4, C3×C4○D4, D4×C9, C22×C18, C3×C4.4D4, D4×C18, C9×C4○D4, C9×C4.4D4
►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 26 46 13)(2 27 47 14)(3 19 48 15)(4 20 49 16)(5 21 50 17)(6 22 51 18)(7 23 52 10)(8 24 53 11)(9 25 54 12)(28 132 39 136)(29 133 40 137)(30 134 41 138)(31 135 42 139)(32 127 43 140)(33 128 44 141)(34 129 45 142)(35 130 37 143)(36 131 38 144)(55 125 68 116)(56 126 69 117)(57 118 70 109)(58 119 71 110)(59 120 72 111)(60 121 64 112)(61 122 65 113)(62 123 66 114)(63 124 67 115)(73 106 82 93)(74 107 83 94)(75 108 84 95)(76 100 85 96)(77 101 86 97)(78 102 87 98)(79 103 88 99)(80 104 89 91)(81 105 90 92)
(1 109 34 106)(2 110 35 107)(3 111 36 108)(4 112 28 100)(5 113 29 101)(6 114 30 102)(7 115 31 103)(8 116 32 104)(9 117 33 105)(10 67 139 79)(11 68 140 80)(12 69 141 81)(13 70 142 73)(14 71 143 74)(15 72 144 75)(16 64 136 76)(17 65 137 77)(18 66 138 78)(19 59 131 84)(20 60 132 85)(21 61 133 86)(22 62 134 87)(23 63 135 88)(24 55 127 89)(25 56 128 90)(26 57 129 82)(27 58 130 83)(37 94 47 119)(38 95 48 120)(39 96 49 121)(40 97 50 122)(41 98 51 123)(42 99 52 124)(43 91 53 125)(44 92 54 126)(45 93 46 118)
(1 93 46 106)(2 94 47 107)(3 95 48 108)(4 96 49 100)(5 97 50 101)(6 98 51 102)(7 99 52 103)(8 91 53 104)(9 92 54 105)(10 79 23 88)(11 80 24 89)(12 81 25 90)(13 73 26 82)(14 74 27 83)(15 75 19 84)(16 76 20 85)(17 77 21 86)(18 78 22 87)(28 121 39 112)(29 122 40 113)(30 123 41 114)(31 124 42 115)(32 125 43 116)(33 126 44 117)(34 118 45 109)(35 119 37 110)(36 120 38 111)(55 140 68 127)(56 141 69 128)(57 142 70 129)(58 143 71 130)(59 144 72 131)(60 136 64 132)(61 137 65 133)(62 138 66 134)(63 139 67 135)
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), (1,26,46,13)(2,27,47,14)(3,19,48,15)(4,20,49,16)(5,21,50,17)(6,22,51,18)(7,23,52,10)(8,24,53,11)(9,25,54,12)(28,132,39,136)(29,133,40,137)(30,134,41,138)(31,135,42,139)(32,127,43,140)(33,128,44,141)(34,129,45,142)(35,130,37,143)(36,131,38,144)(55,125,68,116)(56,126,69,117)(57,118,70,109)(58,119,71,110)(59,120,72,111)(60,121,64,112)(61,122,65,113)(62,123,66,114)(63,124,67,115)(73,106,82,93)(74,107,83,94)(75,108,84,95)(76,100,85,96)(77,101,86,97)(78,102,87,98)(79,103,88,99)(80,104,89,91)(81,105,90,92), (1,109,34,106)(2,110,35,107)(3,111,36,108)(4,112,28,100)(5,113,29,101)(6,114,30,102)(7,115,31,103)(8,116,32,104)(9,117,33,105)(10,67,139,79)(11,68,140,80)(12,69,141,81)(13,70,142,73)(14,71,143,74)(15,72,144,75)(16,64,136,76)(17,65,137,77)(18,66,138,78)(19,59,131,84)(20,60,132,85)(21,61,133,86)(22,62,134,87)(23,63,135,88)(24,55,127,89)(25,56,128,90)(26,57,129,82)(27,58,130,83)(37,94,47,119)(38,95,48,120)(39,96,49,121)(40,97,50,122)(41,98,51,123)(42,99,52,124)(43,91,53,125)(44,92,54,126)(45,93,46,118), (1,93,46,106)(2,94,47,107)(3,95,48,108)(4,96,49,100)(5,97,50,101)(6,98,51,102)(7,99,52,103)(8,91,53,104)(9,92,54,105)(10,79,23,88)(11,80,24,89)(12,81,25,90)(13,73,26,82)(14,74,27,83)(15,75,19,84)(16,76,20,85)(17,77,21,86)(18,78,22,87)(28,121,39,112)(29,122,40,113)(30,123,41,114)(31,124,42,115)(32,125,43,116)(33,126,44,117)(34,118,45,109)(35,119,37,110)(36,120,38,111)(55,140,68,127)(56,141,69,128)(57,142,70,129)(58,143,71,130)(59,144,72,131)(60,136,64,132)(61,137,65,133)(62,138,66,134)(63,139,67,135)>;
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), (1,26,46,13)(2,27,47,14)(3,19,48,15)(4,20,49,16)(5,21,50,17)(6,22,51,18)(7,23,52,10)(8,24,53,11)(9,25,54,12)(28,132,39,136)(29,133,40,137)(30,134,41,138)(31,135,42,139)(32,127,43,140)(33,128,44,141)(34,129,45,142)(35,130,37,143)(36,131,38,144)(55,125,68,116)(56,126,69,117)(57,118,70,109)(58,119,71,110)(59,120,72,111)(60,121,64,112)(61,122,65,113)(62,123,66,114)(63,124,67,115)(73,106,82,93)(74,107,83,94)(75,108,84,95)(76,100,85,96)(77,101,86,97)(78,102,87,98)(79,103,88,99)(80,104,89,91)(81,105,90,92), (1,109,34,106)(2,110,35,107)(3,111,36,108)(4,112,28,100)(5,113,29,101)(6,114,30,102)(7,115,31,103)(8,116,32,104)(9,117,33,105)(10,67,139,79)(11,68,140,80)(12,69,141,81)(13,70,142,73)(14,71,143,74)(15,72,144,75)(16,64,136,76)(17,65,137,77)(18,66,138,78)(19,59,131,84)(20,60,132,85)(21,61,133,86)(22,62,134,87)(23,63,135,88)(24,55,127,89)(25,56,128,90)(26,57,129,82)(27,58,130,83)(37,94,47,119)(38,95,48,120)(39,96,49,121)(40,97,50,122)(41,98,51,123)(42,99,52,124)(43,91,53,125)(44,92,54,126)(45,93,46,118), (1,93,46,106)(2,94,47,107)(3,95,48,108)(4,96,49,100)(5,97,50,101)(6,98,51,102)(7,99,52,103)(8,91,53,104)(9,92,54,105)(10,79,23,88)(11,80,24,89)(12,81,25,90)(13,73,26,82)(14,74,27,83)(15,75,19,84)(16,76,20,85)(17,77,21,86)(18,78,22,87)(28,121,39,112)(29,122,40,113)(30,123,41,114)(31,124,42,115)(32,125,43,116)(33,126,44,117)(34,118,45,109)(35,119,37,110)(36,120,38,111)(55,140,68,127)(56,141,69,128)(57,142,70,129)(58,143,71,130)(59,144,72,131)(60,136,64,132)(61,137,65,133)(62,138,66,134)(63,139,67,135) );
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)], [(1,26,46,13),(2,27,47,14),(3,19,48,15),(4,20,49,16),(5,21,50,17),(6,22,51,18),(7,23,52,10),(8,24,53,11),(9,25,54,12),(28,132,39,136),(29,133,40,137),(30,134,41,138),(31,135,42,139),(32,127,43,140),(33,128,44,141),(34,129,45,142),(35,130,37,143),(36,131,38,144),(55,125,68,116),(56,126,69,117),(57,118,70,109),(58,119,71,110),(59,120,72,111),(60,121,64,112),(61,122,65,113),(62,123,66,114),(63,124,67,115),(73,106,82,93),(74,107,83,94),(75,108,84,95),(76,100,85,96),(77,101,86,97),(78,102,87,98),(79,103,88,99),(80,104,89,91),(81,105,90,92)], [(1,109,34,106),(2,110,35,107),(3,111,36,108),(4,112,28,100),(5,113,29,101),(6,114,30,102),(7,115,31,103),(8,116,32,104),(9,117,33,105),(10,67,139,79),(11,68,140,80),(12,69,141,81),(13,70,142,73),(14,71,143,74),(15,72,144,75),(16,64,136,76),(17,65,137,77),(18,66,138,78),(19,59,131,84),(20,60,132,85),(21,61,133,86),(22,62,134,87),(23,63,135,88),(24,55,127,89),(25,56,128,90),(26,57,129,82),(27,58,130,83),(37,94,47,119),(38,95,48,120),(39,96,49,121),(40,97,50,122),(41,98,51,123),(42,99,52,124),(43,91,53,125),(44,92,54,126),(45,93,46,118)], [(1,93,46,106),(2,94,47,107),(3,95,48,108),(4,96,49,100),(5,97,50,101),(6,98,51,102),(7,99,52,103),(8,91,53,104),(9,92,54,105),(10,79,23,88),(11,80,24,89),(12,81,25,90),(13,73,26,82),(14,74,27,83),(15,75,19,84),(16,76,20,85),(17,77,21,86),(18,78,22,87),(28,121,39,112),(29,122,40,113),(30,123,41,114),(31,124,42,115),(32,125,43,116),(33,126,44,117),(34,118,45,109),(35,119,37,110),(36,120,38,111),(55,140,68,127),(56,141,69,128),(57,142,70,129),(58,143,71,130),(59,144,72,131),(60,136,64,132),(61,137,65,133),(62,138,66,134),(63,139,67,135)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 9A | ··· | 9F | 12A | ··· | 12L | 12M | 12N | 12O | 12P | 18A | ··· | 18R | 18S | ··· | 18AD | 36A | ··· | 36AJ | 36AK | ··· | 36AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
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 |
| type | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | D4×C9 | C9×C4○D4 |
| kernel | C9×C4.4D4 | C4×C36 | C9×C22⋊C4 | D4×C18 | Q8×C18 | C3×C4.4D4 | C4×C12 | C3×C22⋊C4 | C6×D4 | C6×Q8 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C36 | C18 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 24 | 6 | 6 | 2 | 4 | 4 | 8 | 12 | 24 |
Matrix representation of C9×C4.4D4 ►in GL4(𝔽37) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 6 | 0 |
| 2 | 21 | 0 | 0 |
| 35 | 35 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 2 | 21 | 0 | 0 |
| 21 | 35 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 6 |
G:=sub<GL(4,GF(37))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,0,6,0,0,6,0],[2,35,0,0,21,35,0,0,0,0,31,0,0,0,0,31],[2,21,0,0,21,35,0,0,0,0,31,0,0,0,0,6] >;
C9×C4.4D4 in GAP, Magma, Sage, TeX
C_9\times C_4._4D_4
% in TeX
G:=Group("C9xC4.4D4"); // GroupNames label
G:=SmallGroup(288,174);
// by ID
G=gap.SmallGroup(288,174);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,344,1094,142,360]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations