metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.53D4, M4(2).1D9, C22.1Dic18, C9⋊C8.1C4, (C2×C18).Q8, C12.5(C4×S3), C36.5(C2×C4), C4.13(C4×D9), C18.8(C4⋊C4), C9⋊2(C8.C4), (C2×C4).39D18, (C2×C12).41D6, (C2×C6).2Dic6, C4.28(C9⋊D4), C4.Dic9.2C2, C2.5(Dic9⋊C4), (C2×C36).19C22, C3.(C12.53D4), (C3×M4(2)).5S3, (C9×M4(2)).1C2, C12.123(C3⋊D4), C6.13(Dic3⋊C4), (C2×C9⋊C8).4C2, SmallGroup(288,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.53D4
G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=cac-1=a17, cbc-1=a18b3 >
Smallest permutation representation of C36.53D4
►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)
(1 100 28 91 19 82 10 73)(2 81 29 108 20 99 11 90)(3 98 30 89 21 80 12 107)(4 79 31 106 22 97 13 88)(5 96 32 87 23 78 14 105)(6 77 33 104 24 95 15 86)(7 94 34 85 25 76 16 103)(8 75 35 102 26 93 17 84)(9 92 36 83 27 74 18 101)(37 127 46 136 55 109 64 118)(38 144 47 117 56 126 65 135)(39 125 48 134 57 143 66 116)(40 142 49 115 58 124 67 133)(41 123 50 132 59 141 68 114)(42 140 51 113 60 122 69 131)(43 121 52 130 61 139 70 112)(44 138 53 111 62 120 71 129)(45 119 54 128 63 137 72 110)
(1 127 10 136 19 109 28 118)(2 144 11 117 20 126 29 135)(3 125 12 134 21 143 30 116)(4 142 13 115 22 124 31 133)(5 123 14 132 23 141 32 114)(6 140 15 113 24 122 33 131)(7 121 16 130 25 139 34 112)(8 138 17 111 26 120 35 129)(9 119 18 128 27 137 36 110)(37 73 46 82 55 91 64 100)(38 90 47 99 56 108 65 81)(39 107 48 80 57 89 66 98)(40 88 49 97 58 106 67 79)(41 105 50 78 59 87 68 96)(42 86 51 95 60 104 69 77)(43 103 52 76 61 85 70 94)(44 84 53 93 62 102 71 75)(45 101 54 74 63 83 72 92)
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), (1,100,28,91,19,82,10,73)(2,81,29,108,20,99,11,90)(3,98,30,89,21,80,12,107)(4,79,31,106,22,97,13,88)(5,96,32,87,23,78,14,105)(6,77,33,104,24,95,15,86)(7,94,34,85,25,76,16,103)(8,75,35,102,26,93,17,84)(9,92,36,83,27,74,18,101)(37,127,46,136,55,109,64,118)(38,144,47,117,56,126,65,135)(39,125,48,134,57,143,66,116)(40,142,49,115,58,124,67,133)(41,123,50,132,59,141,68,114)(42,140,51,113,60,122,69,131)(43,121,52,130,61,139,70,112)(44,138,53,111,62,120,71,129)(45,119,54,128,63,137,72,110), (1,127,10,136,19,109,28,118)(2,144,11,117,20,126,29,135)(3,125,12,134,21,143,30,116)(4,142,13,115,22,124,31,133)(5,123,14,132,23,141,32,114)(6,140,15,113,24,122,33,131)(7,121,16,130,25,139,34,112)(8,138,17,111,26,120,35,129)(9,119,18,128,27,137,36,110)(37,73,46,82,55,91,64,100)(38,90,47,99,56,108,65,81)(39,107,48,80,57,89,66,98)(40,88,49,97,58,106,67,79)(41,105,50,78,59,87,68,96)(42,86,51,95,60,104,69,77)(43,103,52,76,61,85,70,94)(44,84,53,93,62,102,71,75)(45,101,54,74,63,83,72,92)>;
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), (1,100,28,91,19,82,10,73)(2,81,29,108,20,99,11,90)(3,98,30,89,21,80,12,107)(4,79,31,106,22,97,13,88)(5,96,32,87,23,78,14,105)(6,77,33,104,24,95,15,86)(7,94,34,85,25,76,16,103)(8,75,35,102,26,93,17,84)(9,92,36,83,27,74,18,101)(37,127,46,136,55,109,64,118)(38,144,47,117,56,126,65,135)(39,125,48,134,57,143,66,116)(40,142,49,115,58,124,67,133)(41,123,50,132,59,141,68,114)(42,140,51,113,60,122,69,131)(43,121,52,130,61,139,70,112)(44,138,53,111,62,120,71,129)(45,119,54,128,63,137,72,110), (1,127,10,136,19,109,28,118)(2,144,11,117,20,126,29,135)(3,125,12,134,21,143,30,116)(4,142,13,115,22,124,31,133)(5,123,14,132,23,141,32,114)(6,140,15,113,24,122,33,131)(7,121,16,130,25,139,34,112)(8,138,17,111,26,120,35,129)(9,119,18,128,27,137,36,110)(37,73,46,82,55,91,64,100)(38,90,47,99,56,108,65,81)(39,107,48,80,57,89,66,98)(40,88,49,97,58,106,67,79)(41,105,50,78,59,87,68,96)(42,86,51,95,60,104,69,77)(43,103,52,76,61,85,70,94)(44,84,53,93,62,102,71,75)(45,101,54,74,63,83,72,92) );
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)], [(1,100,28,91,19,82,10,73),(2,81,29,108,20,99,11,90),(3,98,30,89,21,80,12,107),(4,79,31,106,22,97,13,88),(5,96,32,87,23,78,14,105),(6,77,33,104,24,95,15,86),(7,94,34,85,25,76,16,103),(8,75,35,102,26,93,17,84),(9,92,36,83,27,74,18,101),(37,127,46,136,55,109,64,118),(38,144,47,117,56,126,65,135),(39,125,48,134,57,143,66,116),(40,142,49,115,58,124,67,133),(41,123,50,132,59,141,68,114),(42,140,51,113,60,122,69,131),(43,121,52,130,61,139,70,112),(44,138,53,111,62,120,71,129),(45,119,54,128,63,137,72,110)], [(1,127,10,136,19,109,28,118),(2,144,11,117,20,126,29,135),(3,125,12,134,21,143,30,116),(4,142,13,115,22,124,31,133),(5,123,14,132,23,141,32,114),(6,140,15,113,24,122,33,131),(7,121,16,130,25,139,34,112),(8,138,17,111,26,120,35,129),(9,119,18,128,27,137,36,110),(37,73,46,82,55,91,64,100),(38,90,47,99,56,108,65,81),(39,107,48,80,57,89,66,98),(40,88,49,97,58,106,67,79),(41,105,50,78,59,87,68,96),(42,86,51,95,60,104,69,77),(43,103,52,76,61,85,70,94),(44,84,53,93,62,102,71,75),(45,101,54,74,63,83,72,92)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 6 | 6 | 8 | 8 | 8 | 8 | 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 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 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 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | D9 | C4×S3 | C3⋊D4 | Dic6 | C8.C4 | D18 | C4×D9 | C9⋊D4 | Dic18 | C12.53D4 | C36.53D4 |
| kernel | C36.53D4 | C2×C9⋊C8 | C4.Dic9 | C9×M4(2) | C9⋊C8 | C3×M4(2) | C36 | C2×C18 | C2×C12 | M4(2) | C12 | C12 | C2×C6 | C9 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 2 | 4 | 3 | 6 | 6 | 6 | 2 | 6 |
Matrix representation of C36.53D4 ►in GL4(𝔽73) generated by
| 3 | 28 | 0 | 0 |
| 45 | 31 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 59 | 19 | 0 | 0 |
| 5 | 14 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 12 | 51 |
| 51 | 10 | 0 | 0 |
| 61 | 22 | 0 | 0 |
| 0 | 0 | 22 | 51 |
| 0 | 0 | 0 | 51 |
G:=sub<GL(4,GF(73))| [3,45,0,0,28,31,0,0,0,0,46,0,0,0,0,46],[59,5,0,0,19,14,0,0,0,0,63,12,0,0,0,51],[51,61,0,0,10,22,0,0,0,0,22,0,0,0,51,51] >;
C36.53D4 in GAP, Magma, Sage, TeX
C_{36}._{53}D_4 % in TeX
G:=Group("C36.53D4"); // GroupNames label
G:=SmallGroup(288,29);
// by ID
G=gap.SmallGroup(288,29);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^18*b^3>;
// generators/relations
Export