metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic9.1D4, C23.11D18, C22⋊C4⋊5D9, C2.10(D4×D9), C6.82(S3×D4), (C2×C12).4D6, D18⋊C4⋊11C2, C18.21(C2×D4), (C2×C4).28D18, C9⋊2(C4.4D4), (C4×Dic9)⋊12C2, (C2×Dic18)⋊3C2, (C2×C36).5C22, (C22×C6).45D6, C6.80(C4○D12), C18.10(C4○D4), C18.D4⋊5C2, C2.9(D4⋊2D9), (C2×C18).26C23, C6.77(D4⋊2S3), C3.(C23.11D6), C2.12(D36⋊5C2), (C2×Dic9).7C22, (C22×D9).5C22, C22.44(C22×D9), (C22×C18).15C22, (C9×C22⋊C4)⋊7C2, (C2×C9⋊D4).4C2, (C3×C22⋊C4).9S3, (C2×C6).183(C22×S3), SmallGroup(288,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9.D4
G = < a,b,c,d | a18=c4=1, b2=d2=a9, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a9b, dcd-1=a9c-1 >
Subgroups: 524 in 114 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, D9, C18, C18, Dic6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4.4D4, Dic9, Dic9, C36, D18, C2×C18, C2×C18, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, Dic18, C2×Dic9, C9⋊D4, C2×C36, C22×D9, C22×C18, C23.11D6, C4×Dic9, D18⋊C4, C18.D4, C9×C22⋊C4, C2×Dic18, C2×C9⋊D4, Dic9.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C22×S3, C4.4D4, D18, C4○D12, S3×D4, D4⋊2S3, C22×D9, C23.11D6, D36⋊5C2, D4×D9, D4⋊2D9, Dic9.D4
►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)
(1 120 10 111)(2 119 11 110)(3 118 12 109)(4 117 13 126)(5 116 14 125)(6 115 15 124)(7 114 16 123)(8 113 17 122)(9 112 18 121)(19 95 28 104)(20 94 29 103)(21 93 30 102)(22 92 31 101)(23 91 32 100)(24 108 33 99)(25 107 34 98)(26 106 35 97)(27 105 36 96)(37 68 46 59)(38 67 47 58)(39 66 48 57)(40 65 49 56)(41 64 50 55)(42 63 51 72)(43 62 52 71)(44 61 53 70)(45 60 54 69)(73 143 82 134)(74 142 83 133)(75 141 84 132)(76 140 85 131)(77 139 86 130)(78 138 87 129)(79 137 88 128)(80 136 89 127)(81 135 90 144)
(1 36 64 141)(2 19 65 142)(3 20 66 143)(4 21 67 144)(5 22 68 127)(6 23 69 128)(7 24 70 129)(8 25 71 130)(9 26 72 131)(10 27 55 132)(11 28 56 133)(12 29 57 134)(13 30 58 135)(14 31 59 136)(15 32 60 137)(16 33 61 138)(17 34 62 139)(18 35 63 140)(37 89 125 101)(38 90 126 102)(39 73 109 103)(40 74 110 104)(41 75 111 105)(42 76 112 106)(43 77 113 107)(44 78 114 108)(45 79 115 91)(46 80 116 92)(47 81 117 93)(48 82 118 94)(49 83 119 95)(50 84 120 96)(51 85 121 97)(52 86 122 98)(53 87 123 99)(54 88 124 100)
(1 132 10 141)(2 131 11 140)(3 130 12 139)(4 129 13 138)(5 128 14 137)(6 127 15 136)(7 144 16 135)(8 143 17 134)(9 142 18 133)(19 63 28 72)(20 62 29 71)(21 61 30 70)(22 60 31 69)(23 59 32 68)(24 58 33 67)(25 57 34 66)(26 56 35 65)(27 55 36 64)(37 91 46 100)(38 108 47 99)(39 107 48 98)(40 106 49 97)(41 105 50 96)(42 104 51 95)(43 103 52 94)(44 102 53 93)(45 101 54 92)(73 122 82 113)(74 121 83 112)(75 120 84 111)(76 119 85 110)(77 118 86 109)(78 117 87 126)(79 116 88 125)(80 115 89 124)(81 114 90 123)
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), (1,120,10,111)(2,119,11,110)(3,118,12,109)(4,117,13,126)(5,116,14,125)(6,115,15,124)(7,114,16,123)(8,113,17,122)(9,112,18,121)(19,95,28,104)(20,94,29,103)(21,93,30,102)(22,92,31,101)(23,91,32,100)(24,108,33,99)(25,107,34,98)(26,106,35,97)(27,105,36,96)(37,68,46,59)(38,67,47,58)(39,66,48,57)(40,65,49,56)(41,64,50,55)(42,63,51,72)(43,62,52,71)(44,61,53,70)(45,60,54,69)(73,143,82,134)(74,142,83,133)(75,141,84,132)(76,140,85,131)(77,139,86,130)(78,138,87,129)(79,137,88,128)(80,136,89,127)(81,135,90,144), (1,36,64,141)(2,19,65,142)(3,20,66,143)(4,21,67,144)(5,22,68,127)(6,23,69,128)(7,24,70,129)(8,25,71,130)(9,26,72,131)(10,27,55,132)(11,28,56,133)(12,29,57,134)(13,30,58,135)(14,31,59,136)(15,32,60,137)(16,33,61,138)(17,34,62,139)(18,35,63,140)(37,89,125,101)(38,90,126,102)(39,73,109,103)(40,74,110,104)(41,75,111,105)(42,76,112,106)(43,77,113,107)(44,78,114,108)(45,79,115,91)(46,80,116,92)(47,81,117,93)(48,82,118,94)(49,83,119,95)(50,84,120,96)(51,85,121,97)(52,86,122,98)(53,87,123,99)(54,88,124,100), (1,132,10,141)(2,131,11,140)(3,130,12,139)(4,129,13,138)(5,128,14,137)(6,127,15,136)(7,144,16,135)(8,143,17,134)(9,142,18,133)(19,63,28,72)(20,62,29,71)(21,61,30,70)(22,60,31,69)(23,59,32,68)(24,58,33,67)(25,57,34,66)(26,56,35,65)(27,55,36,64)(37,91,46,100)(38,108,47,99)(39,107,48,98)(40,106,49,97)(41,105,50,96)(42,104,51,95)(43,103,52,94)(44,102,53,93)(45,101,54,92)(73,122,82,113)(74,121,83,112)(75,120,84,111)(76,119,85,110)(77,118,86,109)(78,117,87,126)(79,116,88,125)(80,115,89,124)(81,114,90,123)>;
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), (1,120,10,111)(2,119,11,110)(3,118,12,109)(4,117,13,126)(5,116,14,125)(6,115,15,124)(7,114,16,123)(8,113,17,122)(9,112,18,121)(19,95,28,104)(20,94,29,103)(21,93,30,102)(22,92,31,101)(23,91,32,100)(24,108,33,99)(25,107,34,98)(26,106,35,97)(27,105,36,96)(37,68,46,59)(38,67,47,58)(39,66,48,57)(40,65,49,56)(41,64,50,55)(42,63,51,72)(43,62,52,71)(44,61,53,70)(45,60,54,69)(73,143,82,134)(74,142,83,133)(75,141,84,132)(76,140,85,131)(77,139,86,130)(78,138,87,129)(79,137,88,128)(80,136,89,127)(81,135,90,144), (1,36,64,141)(2,19,65,142)(3,20,66,143)(4,21,67,144)(5,22,68,127)(6,23,69,128)(7,24,70,129)(8,25,71,130)(9,26,72,131)(10,27,55,132)(11,28,56,133)(12,29,57,134)(13,30,58,135)(14,31,59,136)(15,32,60,137)(16,33,61,138)(17,34,62,139)(18,35,63,140)(37,89,125,101)(38,90,126,102)(39,73,109,103)(40,74,110,104)(41,75,111,105)(42,76,112,106)(43,77,113,107)(44,78,114,108)(45,79,115,91)(46,80,116,92)(47,81,117,93)(48,82,118,94)(49,83,119,95)(50,84,120,96)(51,85,121,97)(52,86,122,98)(53,87,123,99)(54,88,124,100), (1,132,10,141)(2,131,11,140)(3,130,12,139)(4,129,13,138)(5,128,14,137)(6,127,15,136)(7,144,16,135)(8,143,17,134)(9,142,18,133)(19,63,28,72)(20,62,29,71)(21,61,30,70)(22,60,31,69)(23,59,32,68)(24,58,33,67)(25,57,34,66)(26,56,35,65)(27,55,36,64)(37,91,46,100)(38,108,47,99)(39,107,48,98)(40,106,49,97)(41,105,50,96)(42,104,51,95)(43,103,52,94)(44,102,53,93)(45,101,54,92)(73,122,82,113)(74,121,83,112)(75,120,84,111)(76,119,85,110)(77,118,86,109)(78,117,87,126)(79,116,88,125)(80,115,89,124)(81,114,90,123) );
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)], [(1,120,10,111),(2,119,11,110),(3,118,12,109),(4,117,13,126),(5,116,14,125),(6,115,15,124),(7,114,16,123),(8,113,17,122),(9,112,18,121),(19,95,28,104),(20,94,29,103),(21,93,30,102),(22,92,31,101),(23,91,32,100),(24,108,33,99),(25,107,34,98),(26,106,35,97),(27,105,36,96),(37,68,46,59),(38,67,47,58),(39,66,48,57),(40,65,49,56),(41,64,50,55),(42,63,51,72),(43,62,52,71),(44,61,53,70),(45,60,54,69),(73,143,82,134),(74,142,83,133),(75,141,84,132),(76,140,85,131),(77,139,86,130),(78,138,87,129),(79,137,88,128),(80,136,89,127),(81,135,90,144)], [(1,36,64,141),(2,19,65,142),(3,20,66,143),(4,21,67,144),(5,22,68,127),(6,23,69,128),(7,24,70,129),(8,25,71,130),(9,26,72,131),(10,27,55,132),(11,28,56,133),(12,29,57,134),(13,30,58,135),(14,31,59,136),(15,32,60,137),(16,33,61,138),(17,34,62,139),(18,35,63,140),(37,89,125,101),(38,90,126,102),(39,73,109,103),(40,74,110,104),(41,75,111,105),(42,76,112,106),(43,77,113,107),(44,78,114,108),(45,79,115,91),(46,80,116,92),(47,81,117,93),(48,82,118,94),(49,83,119,95),(50,84,120,96),(51,85,121,97),(52,86,122,98),(53,87,123,99),(54,88,124,100)], [(1,132,10,141),(2,131,11,140),(3,130,12,139),(4,129,13,138),(5,128,14,137),(6,127,15,136),(7,144,16,135),(8,143,17,134),(9,142,18,133),(19,63,28,72),(20,62,29,71),(21,61,30,70),(22,60,31,69),(23,59,32,68),(24,58,33,67),(25,57,34,66),(26,56,35,65),(27,55,36,64),(37,91,46,100),(38,108,47,99),(39,107,48,98),(40,106,49,97),(41,105,50,96),(42,104,51,95),(43,103,52,94),(44,102,53,93),(45,101,54,92),(73,122,82,113),(74,121,83,112),(75,120,84,111),(76,119,85,110),(77,118,86,109),(78,117,87,126),(79,116,88,125),(80,115,89,124),(81,114,90,123)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 36 | 2 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D9 | D18 | D18 | C4○D12 | D36⋊5C2 | S3×D4 | D4⋊2S3 | D4×D9 | D4⋊2D9 |
| kernel | Dic9.D4 | C4×Dic9 | D18⋊C4 | C18.D4 | C9×C22⋊C4 | C2×Dic18 | C2×C9⋊D4 | C3×C22⋊C4 | Dic9 | C2×C12 | C22×C6 | C18 | C22⋊C4 | C2×C4 | C23 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 3 | 6 | 3 | 4 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of Dic9.D4 ►in GL6(𝔽37)
| 17 | 11 | 0 | 0 | 0 | 0 |
| 26 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 35 | 36 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 25 | 31 |
G:=sub<GL(6,GF(37))| [17,26,0,0,0,0,11,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,36,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,36,0,0,0,0,0,0,0,1,35,0,0,0,0,1,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,31,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,6,0,0,0,0,0,0,0,6,25,0,0,0,0,0,31] >;
Dic9.D4 in GAP, Magma, Sage, TeX
{\rm Dic}_9.D_4 % in TeX
G:=Group("Dic9.D4"); // GroupNames label
G:=SmallGroup(288,95);
// by ID
G=gap.SmallGroup(288,95);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^18=c^4=1,b^2=d^2=a^9,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=a^9*c^-1>;
// generators/relations