metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.47D4, C12.3D12, C4.11D36, M4(2).2D9, (C2×C4).1D18, (C2×Dic9).C4, (C2×C12).42D6, C22.4(C4×D9), C6.15(D6⋊C4), C2.9(D18⋊C4), C4.21(C9⋊D4), C9⋊1(C4.10D4), C4.Dic9.3C2, C18.8(C22⋊C4), (C2×C36).20C22, C3.(C12.47D4), (C2×Dic18).6C2, (C3×M4(2)).6S3, (C9×M4(2)).2C2, C12.108(C3⋊D4), (C2×C6).3(C4×S3), (C2×C18).2(C2×C4), SmallGroup(288,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.47D4
G = < a,b,c | a36=1, b4=c2=a18, bab-1=cac-1=a-1, cbc-1=a9b3 >
Subgroups: 228 in 57 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C9, Dic3 [×2], C12 [×2], C2×C6, M4(2), M4(2), C2×Q8, C18, C18, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12, C4.10D4, Dic9 [×2], C36 [×2], C2×C18, C4.Dic3, C3×M4(2), C2×Dic6, C9⋊C8, C72, Dic18 [×2], C2×Dic9 [×2], C2×C36, C12.47D4, C4.Dic9, C9×M4(2), C2×Dic18, C36.47D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4.10D4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C12.47D4, D18⋊C4, C36.47D4
►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)
(1 136 10 127 19 118 28 109)(2 135 11 126 20 117 29 144)(3 134 12 125 21 116 30 143)(4 133 13 124 22 115 31 142)(5 132 14 123 23 114 32 141)(6 131 15 122 24 113 33 140)(7 130 16 121 25 112 34 139)(8 129 17 120 26 111 35 138)(9 128 18 119 27 110 36 137)(37 96 64 105 55 78 46 87)(38 95 65 104 56 77 47 86)(39 94 66 103 57 76 48 85)(40 93 67 102 58 75 49 84)(41 92 68 101 59 74 50 83)(42 91 69 100 60 73 51 82)(43 90 70 99 61 108 52 81)(44 89 71 98 62 107 53 80)(45 88 72 97 63 106 54 79)
(1 78 19 96)(2 77 20 95)(3 76 21 94)(4 75 22 93)(5 74 23 92)(6 73 24 91)(7 108 25 90)(8 107 26 89)(9 106 27 88)(10 105 28 87)(11 104 29 86)(12 103 30 85)(13 102 31 84)(14 101 32 83)(15 100 33 82)(16 99 34 81)(17 98 35 80)(18 97 36 79)(37 109 55 127)(38 144 56 126)(39 143 57 125)(40 142 58 124)(41 141 59 123)(42 140 60 122)(43 139 61 121)(44 138 62 120)(45 137 63 119)(46 136 64 118)(47 135 65 117)(48 134 66 116)(49 133 67 115)(50 132 68 114)(51 131 69 113)(52 130 70 112)(53 129 71 111)(54 128 72 110)
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), (1,136,10,127,19,118,28,109)(2,135,11,126,20,117,29,144)(3,134,12,125,21,116,30,143)(4,133,13,124,22,115,31,142)(5,132,14,123,23,114,32,141)(6,131,15,122,24,113,33,140)(7,130,16,121,25,112,34,139)(8,129,17,120,26,111,35,138)(9,128,18,119,27,110,36,137)(37,96,64,105,55,78,46,87)(38,95,65,104,56,77,47,86)(39,94,66,103,57,76,48,85)(40,93,67,102,58,75,49,84)(41,92,68,101,59,74,50,83)(42,91,69,100,60,73,51,82)(43,90,70,99,61,108,52,81)(44,89,71,98,62,107,53,80)(45,88,72,97,63,106,54,79), (1,78,19,96)(2,77,20,95)(3,76,21,94)(4,75,22,93)(5,74,23,92)(6,73,24,91)(7,108,25,90)(8,107,26,89)(9,106,27,88)(10,105,28,87)(11,104,29,86)(12,103,30,85)(13,102,31,84)(14,101,32,83)(15,100,33,82)(16,99,34,81)(17,98,35,80)(18,97,36,79)(37,109,55,127)(38,144,56,126)(39,143,57,125)(40,142,58,124)(41,141,59,123)(42,140,60,122)(43,139,61,121)(44,138,62,120)(45,137,63,119)(46,136,64,118)(47,135,65,117)(48,134,66,116)(49,133,67,115)(50,132,68,114)(51,131,69,113)(52,130,70,112)(53,129,71,111)(54,128,72,110)>;
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), (1,136,10,127,19,118,28,109)(2,135,11,126,20,117,29,144)(3,134,12,125,21,116,30,143)(4,133,13,124,22,115,31,142)(5,132,14,123,23,114,32,141)(6,131,15,122,24,113,33,140)(7,130,16,121,25,112,34,139)(8,129,17,120,26,111,35,138)(9,128,18,119,27,110,36,137)(37,96,64,105,55,78,46,87)(38,95,65,104,56,77,47,86)(39,94,66,103,57,76,48,85)(40,93,67,102,58,75,49,84)(41,92,68,101,59,74,50,83)(42,91,69,100,60,73,51,82)(43,90,70,99,61,108,52,81)(44,89,71,98,62,107,53,80)(45,88,72,97,63,106,54,79), (1,78,19,96)(2,77,20,95)(3,76,21,94)(4,75,22,93)(5,74,23,92)(6,73,24,91)(7,108,25,90)(8,107,26,89)(9,106,27,88)(10,105,28,87)(11,104,29,86)(12,103,30,85)(13,102,31,84)(14,101,32,83)(15,100,33,82)(16,99,34,81)(17,98,35,80)(18,97,36,79)(37,109,55,127)(38,144,56,126)(39,143,57,125)(40,142,58,124)(41,141,59,123)(42,140,60,122)(43,139,61,121)(44,138,62,120)(45,137,63,119)(46,136,64,118)(47,135,65,117)(48,134,66,116)(49,133,67,115)(50,132,68,114)(51,131,69,113)(52,130,70,112)(53,129,71,111)(54,128,72,110) );
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)], [(1,136,10,127,19,118,28,109),(2,135,11,126,20,117,29,144),(3,134,12,125,21,116,30,143),(4,133,13,124,22,115,31,142),(5,132,14,123,23,114,32,141),(6,131,15,122,24,113,33,140),(7,130,16,121,25,112,34,139),(8,129,17,120,26,111,35,138),(9,128,18,119,27,110,36,137),(37,96,64,105,55,78,46,87),(38,95,65,104,56,77,47,86),(39,94,66,103,57,76,48,85),(40,93,67,102,58,75,49,84),(41,92,68,101,59,74,50,83),(42,91,69,100,60,73,51,82),(43,90,70,99,61,108,52,81),(44,89,71,98,62,107,53,80),(45,88,72,97,63,106,54,79)], [(1,78,19,96),(2,77,20,95),(3,76,21,94),(4,75,22,93),(5,74,23,92),(6,73,24,91),(7,108,25,90),(8,107,26,89),(9,106,27,88),(10,105,28,87),(11,104,29,86),(12,103,30,85),(13,102,31,84),(14,101,32,83),(15,100,33,82),(16,99,34,81),(17,98,35,80),(18,97,36,79),(37,109,55,127),(38,144,56,126),(39,143,57,125),(40,142,58,124),(41,141,59,123),(42,140,60,122),(43,139,61,121),(44,138,62,120),(45,137,63,119),(46,136,64,118),(47,135,65,117),(48,134,66,116),(49,133,67,115),(50,132,68,114),(51,131,69,113),(52,130,70,112),(53,129,71,111),(54,128,72,110)])
51 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | 36H | 36I | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D9 | D12 | C3⋊D4 | C4×S3 | D18 | D36 | C9⋊D4 | C4×D9 | C4.10D4 | C12.47D4 | C36.47D4 |
| kernel | C36.47D4 | C4.Dic9 | C9×M4(2) | C2×Dic18 | C2×Dic9 | C3×M4(2) | C36 | C2×C12 | M4(2) | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of C36.47D4 ►in GL4(𝔽73) generated by
| 25 | 19 | 0 | 0 |
| 54 | 44 | 0 | 0 |
| 0 | 0 | 25 | 19 |
| 0 | 0 | 54 | 44 |
| 0 | 0 | 14 | 5 |
| 0 | 0 | 19 | 59 |
| 51 | 12 | 0 | 0 |
| 63 | 22 | 0 | 0 |
| 14 | 5 | 0 | 0 |
| 19 | 59 | 0 | 0 |
| 0 | 0 | 14 | 5 |
| 0 | 0 | 19 | 59 |
G:=sub<GL(4,GF(73))| [25,54,0,0,19,44,0,0,0,0,25,54,0,0,19,44],[0,0,51,63,0,0,12,22,14,19,0,0,5,59,0,0],[14,19,0,0,5,59,0,0,0,0,14,19,0,0,5,59] >;
C36.47D4 in GAP, Magma, Sage, TeX
C_{36}._{47}D_4 % in TeX
G:=Group("C36.47D4"); // GroupNames label
G:=SmallGroup(288,30);
// by ID
G=gap.SmallGroup(288,30);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^4=c^2=a^18,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^3>;
// generators/relations