metabelian, supersoluble, monomial
Aliases: C36.39D6, D18.Dic3, C12.39D18, Dic9.Dic3, C3⋊C8⋊4D9, C12.61S32, (C4×D9).2S3, (C6×D9).2C4, C4.24(S3×D9), C6.17(C4×D9), (C3×C9)⋊1M4(2), C3⋊3(C8⋊D9), (C12×D9).4C2, C2.3(Dic3×D9), C6.2(S3×Dic3), C36.S3⋊8C2, (C3×C12).157D6, C9⋊1(C4.Dic3), (C3×Dic9).2C4, C18.2(C2×Dic3), (C3×C36).38C22, C32.2(C8⋊S3), C3.1(D6.Dic3), (C9×C3⋊C8)⋊6C2, (C3×C3⋊C8).9S3, (C3×C6).35(C4×S3), (C3×C18).3(C2×C4), SmallGroup(432,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.39D6
G = < a,b,c | a36=b6=1, c2=a27, bab-1=cac-1=a17, cbc-1=a18b-1 >
Subgroups: 300 in 68 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×C12, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C4.Dic3, C3×D9, C3×C18, C9⋊C8, C72, C4×D9, C3×C3⋊C8, C32⋊4C8, S3×C12, C3×Dic9, C3×C36, C6×D9, C8⋊D9, D6.Dic3, C9×C3⋊C8, C36.S3, C12×D9, C36.39D6
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), D9, C4×S3, C2×Dic3, D18, S32, C8⋊S3, C4.Dic3, C4×D9, S3×Dic3, S3×D9, C8⋊D9, D6.Dic3, Dic3×D9, C36.39D6
►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 86 13 74 25 98)(2 103 14 91 26 79)(3 84 15 108 27 96)(4 101 16 89 28 77)(5 82 17 106 29 94)(6 99 18 87 30 75)(7 80 19 104 31 92)(8 97 20 85 32 73)(9 78 21 102 33 90)(10 95 22 83 34 107)(11 76 23 100 35 88)(12 93 24 81 36 105)(37 118 61 130 49 142)(38 135 62 111 50 123)(39 116 63 128 51 140)(40 133 64 109 52 121)(41 114 65 126 53 138)(42 131 66 143 54 119)(43 112 67 124 55 136)(44 129 68 141 56 117)(45 110 69 122 57 134)(46 127 70 139 58 115)(47 144 71 120 59 132)(48 125 72 137 60 113)
(1 144 28 135 19 126 10 117)(2 125 29 116 20 143 11 134)(3 142 30 133 21 124 12 115)(4 123 31 114 22 141 13 132)(5 140 32 131 23 122 14 113)(6 121 33 112 24 139 15 130)(7 138 34 129 25 120 16 111)(8 119 35 110 26 137 17 128)(9 136 36 127 27 118 18 109)(37 105 64 96 55 87 46 78)(38 86 65 77 56 104 47 95)(39 103 66 94 57 85 48 76)(40 84 67 75 58 102 49 93)(41 101 68 92 59 83 50 74)(42 82 69 73 60 100 51 91)(43 99 70 90 61 81 52 108)(44 80 71 107 62 98 53 89)(45 97 72 88 63 79 54 106)
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), (1,86,13,74,25,98)(2,103,14,91,26,79)(3,84,15,108,27,96)(4,101,16,89,28,77)(5,82,17,106,29,94)(6,99,18,87,30,75)(7,80,19,104,31,92)(8,97,20,85,32,73)(9,78,21,102,33,90)(10,95,22,83,34,107)(11,76,23,100,35,88)(12,93,24,81,36,105)(37,118,61,130,49,142)(38,135,62,111,50,123)(39,116,63,128,51,140)(40,133,64,109,52,121)(41,114,65,126,53,138)(42,131,66,143,54,119)(43,112,67,124,55,136)(44,129,68,141,56,117)(45,110,69,122,57,134)(46,127,70,139,58,115)(47,144,71,120,59,132)(48,125,72,137,60,113), (1,144,28,135,19,126,10,117)(2,125,29,116,20,143,11,134)(3,142,30,133,21,124,12,115)(4,123,31,114,22,141,13,132)(5,140,32,131,23,122,14,113)(6,121,33,112,24,139,15,130)(7,138,34,129,25,120,16,111)(8,119,35,110,26,137,17,128)(9,136,36,127,27,118,18,109)(37,105,64,96,55,87,46,78)(38,86,65,77,56,104,47,95)(39,103,66,94,57,85,48,76)(40,84,67,75,58,102,49,93)(41,101,68,92,59,83,50,74)(42,82,69,73,60,100,51,91)(43,99,70,90,61,81,52,108)(44,80,71,107,62,98,53,89)(45,97,72,88,63,79,54,106)>;
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), (1,86,13,74,25,98)(2,103,14,91,26,79)(3,84,15,108,27,96)(4,101,16,89,28,77)(5,82,17,106,29,94)(6,99,18,87,30,75)(7,80,19,104,31,92)(8,97,20,85,32,73)(9,78,21,102,33,90)(10,95,22,83,34,107)(11,76,23,100,35,88)(12,93,24,81,36,105)(37,118,61,130,49,142)(38,135,62,111,50,123)(39,116,63,128,51,140)(40,133,64,109,52,121)(41,114,65,126,53,138)(42,131,66,143,54,119)(43,112,67,124,55,136)(44,129,68,141,56,117)(45,110,69,122,57,134)(46,127,70,139,58,115)(47,144,71,120,59,132)(48,125,72,137,60,113), (1,144,28,135,19,126,10,117)(2,125,29,116,20,143,11,134)(3,142,30,133,21,124,12,115)(4,123,31,114,22,141,13,132)(5,140,32,131,23,122,14,113)(6,121,33,112,24,139,15,130)(7,138,34,129,25,120,16,111)(8,119,35,110,26,137,17,128)(9,136,36,127,27,118,18,109)(37,105,64,96,55,87,46,78)(38,86,65,77,56,104,47,95)(39,103,66,94,57,85,48,76)(40,84,67,75,58,102,49,93)(41,101,68,92,59,83,50,74)(42,82,69,73,60,100,51,91)(43,99,70,90,61,81,52,108)(44,80,71,107,62,98,53,89)(45,97,72,88,63,79,54,106) );
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)], [(1,86,13,74,25,98),(2,103,14,91,26,79),(3,84,15,108,27,96),(4,101,16,89,28,77),(5,82,17,106,29,94),(6,99,18,87,30,75),(7,80,19,104,31,92),(8,97,20,85,32,73),(9,78,21,102,33,90),(10,95,22,83,34,107),(11,76,23,100,35,88),(12,93,24,81,36,105),(37,118,61,130,49,142),(38,135,62,111,50,123),(39,116,63,128,51,140),(40,133,64,109,52,121),(41,114,65,126,53,138),(42,131,66,143,54,119),(43,112,67,124,55,136),(44,129,68,141,56,117),(45,110,69,122,57,134),(46,127,70,139,58,115),(47,144,71,120,59,132),(48,125,72,137,60,113)], [(1,144,28,135,19,126,10,117),(2,125,29,116,20,143,11,134),(3,142,30,133,21,124,12,115),(4,123,31,114,22,141,13,132),(5,140,32,131,23,122,14,113),(6,121,33,112,24,139,15,130),(7,138,34,129,25,120,16,111),(8,119,35,110,26,137,17,128),(9,136,36,127,27,118,18,109),(37,105,64,96,55,87,46,78),(38,86,65,77,56,104,47,95),(39,103,66,94,57,85,48,76),(40,84,67,75,58,102,49,93),(41,101,68,92,59,83,50,74),(42,82,69,73,60,100,51,91),(43,99,70,90,61,81,52,108),(44,80,71,107,62,98,53,89),(45,97,72,88,63,79,54,106)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 18 | 2 | 2 | 4 | 1 | 1 | 18 | 2 | 2 | 4 | 18 | 18 | 6 | 6 | 54 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | S3 | Dic3 | D6 | Dic3 | D6 | M4(2) | D9 | C4×S3 | D18 | C4.Dic3 | C8⋊S3 | C4×D9 | C8⋊D9 | S32 | S3×Dic3 | S3×D9 | D6.Dic3 | Dic3×D9 | C36.39D6 |
| kernel | C36.39D6 | C9×C3⋊C8 | C36.S3 | C12×D9 | C3×Dic9 | C6×D9 | C4×D9 | C3×C3⋊C8 | Dic9 | C36 | D18 | C3×C12 | C3×C9 | C3⋊C8 | C3×C6 | C12 | C9 | C32 | C6 | C3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 4 | 6 | 12 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of C36.39D6 ►in GL6(𝔽73)
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 28 |
| 0 | 0 | 0 | 0 | 45 | 31 |
| 57 | 59 | 0 | 0 | 0 | 0 |
| 13 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 70 | 45 |
| 0 | 0 | 0 | 0 | 42 | 3 |
| 60 | 57 | 0 | 0 | 0 | 0 |
| 67 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 47 |
| 0 | 0 | 0 | 0 | 39 | 8 |
G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,3,45,0,0,0,0,28,31],[57,13,0,0,0,0,59,16,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,70,42,0,0,0,0,45,3],[60,67,0,0,0,0,57,13,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,65,39,0,0,0,0,47,8] >;
C36.39D6 in GAP, Magma, Sage, TeX
C_{36}._{39}D_6 % in TeX
G:=Group("C36.39D6"); // GroupNames label
G:=SmallGroup(432,60);
// by ID
G=gap.SmallGroup(432,60);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,571,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=b^6=1,c^2=a^27,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^18*b^-1>;
// generators/relations