metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.17D4, C23.12D18, (C6×D4).7S3, (C2×D4).6D9, (C4×Dic9)⋊5C2, (D4×C18).5C2, C18.49(C2×D4), (C2×C12).59D6, (C2×C4).51D18, C4.7(C9⋊D4), C9⋊3(C4.4D4), (C22×C6).49D6, (C2×Dic18)⋊10C2, C18.30(C4○D4), C18.D4⋊9C2, C12.12(C3⋊D4), (C2×C18).51C23, (C2×C36).37C22, C3.(C23.12D6), C6.87(D4⋊2S3), C2.16(D4⋊2D9), C22.58(C22×D9), (C22×C18).19C22, (C2×Dic9).16C22, C6.96(C2×C3⋊D4), C2.12(C2×C9⋊D4), (C2×C6).208(C22×S3), SmallGroup(288,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.17D4
G = < a,b,c | a36=b4=1, c2=a18, bab-1=a17, cac-1=a-1, cbc-1=a18b-1 >
Subgroups: 404 in 114 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C9, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C18, C18 [×2], C18 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.4D4, Dic9 [×4], C36 [×2], C2×C18, C2×C18 [×6], C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, Dic18 [×2], C2×Dic9 [×4], C2×C36, D4×C9 [×2], C22×C18 [×2], C23.12D6, C4×Dic9, C18.D4 [×4], C2×Dic18, D4×C18, C36.17D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C3⋊D4 [×2], C22×S3, C4.4D4, D18 [×3], D4⋊2S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C23.12D6, D4⋊2D9 [×2], C2×C9⋊D4, C36.17D4
►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)
(1 105 39 142)(2 86 40 123)(3 103 41 140)(4 84 42 121)(5 101 43 138)(6 82 44 119)(7 99 45 136)(8 80 46 117)(9 97 47 134)(10 78 48 115)(11 95 49 132)(12 76 50 113)(13 93 51 130)(14 74 52 111)(15 91 53 128)(16 108 54 109)(17 89 55 126)(18 106 56 143)(19 87 57 124)(20 104 58 141)(21 85 59 122)(22 102 60 139)(23 83 61 120)(24 100 62 137)(25 81 63 118)(26 98 64 135)(27 79 65 116)(28 96 66 133)(29 77 67 114)(30 94 68 131)(31 75 69 112)(32 92 70 129)(33 73 71 110)(34 90 72 127)(35 107 37 144)(36 88 38 125)
(1 133 19 115)(2 132 20 114)(3 131 21 113)(4 130 22 112)(5 129 23 111)(6 128 24 110)(7 127 25 109)(8 126 26 144)(9 125 27 143)(10 124 28 142)(11 123 29 141)(12 122 30 140)(13 121 31 139)(14 120 32 138)(15 119 33 137)(16 118 34 136)(17 117 35 135)(18 116 36 134)(37 98 55 80)(38 97 56 79)(39 96 57 78)(40 95 58 77)(41 94 59 76)(42 93 60 75)(43 92 61 74)(44 91 62 73)(45 90 63 108)(46 89 64 107)(47 88 65 106)(48 87 66 105)(49 86 67 104)(50 85 68 103)(51 84 69 102)(52 83 70 101)(53 82 71 100)(54 81 72 99)
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), (1,105,39,142)(2,86,40,123)(3,103,41,140)(4,84,42,121)(5,101,43,138)(6,82,44,119)(7,99,45,136)(8,80,46,117)(9,97,47,134)(10,78,48,115)(11,95,49,132)(12,76,50,113)(13,93,51,130)(14,74,52,111)(15,91,53,128)(16,108,54,109)(17,89,55,126)(18,106,56,143)(19,87,57,124)(20,104,58,141)(21,85,59,122)(22,102,60,139)(23,83,61,120)(24,100,62,137)(25,81,63,118)(26,98,64,135)(27,79,65,116)(28,96,66,133)(29,77,67,114)(30,94,68,131)(31,75,69,112)(32,92,70,129)(33,73,71,110)(34,90,72,127)(35,107,37,144)(36,88,38,125), (1,133,19,115)(2,132,20,114)(3,131,21,113)(4,130,22,112)(5,129,23,111)(6,128,24,110)(7,127,25,109)(8,126,26,144)(9,125,27,143)(10,124,28,142)(11,123,29,141)(12,122,30,140)(13,121,31,139)(14,120,32,138)(15,119,33,137)(16,118,34,136)(17,117,35,135)(18,116,36,134)(37,98,55,80)(38,97,56,79)(39,96,57,78)(40,95,58,77)(41,94,59,76)(42,93,60,75)(43,92,61,74)(44,91,62,73)(45,90,63,108)(46,89,64,107)(47,88,65,106)(48,87,66,105)(49,86,67,104)(50,85,68,103)(51,84,69,102)(52,83,70,101)(53,82,71,100)(54,81,72,99)>;
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), (1,105,39,142)(2,86,40,123)(3,103,41,140)(4,84,42,121)(5,101,43,138)(6,82,44,119)(7,99,45,136)(8,80,46,117)(9,97,47,134)(10,78,48,115)(11,95,49,132)(12,76,50,113)(13,93,51,130)(14,74,52,111)(15,91,53,128)(16,108,54,109)(17,89,55,126)(18,106,56,143)(19,87,57,124)(20,104,58,141)(21,85,59,122)(22,102,60,139)(23,83,61,120)(24,100,62,137)(25,81,63,118)(26,98,64,135)(27,79,65,116)(28,96,66,133)(29,77,67,114)(30,94,68,131)(31,75,69,112)(32,92,70,129)(33,73,71,110)(34,90,72,127)(35,107,37,144)(36,88,38,125), (1,133,19,115)(2,132,20,114)(3,131,21,113)(4,130,22,112)(5,129,23,111)(6,128,24,110)(7,127,25,109)(8,126,26,144)(9,125,27,143)(10,124,28,142)(11,123,29,141)(12,122,30,140)(13,121,31,139)(14,120,32,138)(15,119,33,137)(16,118,34,136)(17,117,35,135)(18,116,36,134)(37,98,55,80)(38,97,56,79)(39,96,57,78)(40,95,58,77)(41,94,59,76)(42,93,60,75)(43,92,61,74)(44,91,62,73)(45,90,63,108)(46,89,64,107)(47,88,65,106)(48,87,66,105)(49,86,67,104)(50,85,68,103)(51,84,69,102)(52,83,70,101)(53,82,71,100)(54,81,72,99) );
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)], [(1,105,39,142),(2,86,40,123),(3,103,41,140),(4,84,42,121),(5,101,43,138),(6,82,44,119),(7,99,45,136),(8,80,46,117),(9,97,47,134),(10,78,48,115),(11,95,49,132),(12,76,50,113),(13,93,51,130),(14,74,52,111),(15,91,53,128),(16,108,54,109),(17,89,55,126),(18,106,56,143),(19,87,57,124),(20,104,58,141),(21,85,59,122),(22,102,60,139),(23,83,61,120),(24,100,62,137),(25,81,63,118),(26,98,64,135),(27,79,65,116),(28,96,66,133),(29,77,67,114),(30,94,68,131),(31,75,69,112),(32,92,70,129),(33,73,71,110),(34,90,72,127),(35,107,37,144),(36,88,38,125)], [(1,133,19,115),(2,132,20,114),(3,131,21,113),(4,130,22,112),(5,129,23,111),(6,128,24,110),(7,127,25,109),(8,126,26,144),(9,125,27,143),(10,124,28,142),(11,123,29,141),(12,122,30,140),(13,121,31,139),(14,120,32,138),(15,119,33,137),(16,118,34,136),(17,117,35,135),(18,116,36,134),(37,98,55,80),(38,97,56,79),(39,96,57,78),(40,95,58,77),(41,94,59,76),(42,93,60,75),(43,92,61,74),(44,91,62,73),(45,90,63,108),(46,89,64,107),(47,88,65,106),(48,87,66,105),(49,86,67,104),(50,85,68,103),(51,84,69,102),(52,83,70,101),(53,82,71,100),(54,81,72,99)])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D9 | C3⋊D4 | D18 | D18 | C9⋊D4 | D4⋊2S3 | D4⋊2D9 |
| kernel | C36.17D4 | C4×Dic9 | C18.D4 | C2×Dic18 | D4×C18 | C6×D4 | C36 | C2×C12 | C22×C6 | C18 | C2×D4 | C12 | C2×C4 | C23 | C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 3 | 4 | 3 | 6 | 12 | 2 | 6 |
Matrix representation of C36.17D4 ►in GL4(𝔽37) generated by
| 0 | 36 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 25 | 26 |
| 0 | 0 | 0 | 3 |
| 0 | 6 | 0 | 0 |
| 31 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 12 | 31 |
| 31 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 31 |
| 0 | 0 | 12 | 31 |
G:=sub<GL(4,GF(37))| [0,1,0,0,36,0,0,0,0,0,25,0,0,0,26,3],[0,31,0,0,6,0,0,0,0,0,6,12,0,0,0,31],[31,0,0,0,0,6,0,0,0,0,6,12,0,0,31,31] >;
C36.17D4 in GAP, Magma, Sage, TeX
C_{36}._{17}D_4 % in TeX
G:=Group("C36.17D4"); // GroupNames label
G:=SmallGroup(288,146);
// by ID
G=gap.SmallGroup(288,146);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,135,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=1,c^2=a^18,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=a^18*b^-1>;
// generators/relations