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, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C9, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C18, C18, C18, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C4.4D4, Dic9, C36, C2×C18, C2×C18, C4×Dic3, C6.D4, C2×Dic6, C6×D4, Dic18, C2×Dic9, C2×C36, D4×C9, C22×C18, C23.12D6, C4×Dic9, C18.D4, C2×Dic18, D4×C18, C36.17D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C3⋊D4, C22×S3, C4.4D4, D18, D4⋊2S3, C2×C3⋊D4, C9⋊D4, C22×D9, C23.12D6, D4⋊2D9, 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)
(1 131 79 57)(2 112 80 38)(3 129 81 55)(4 110 82 72)(5 127 83 53)(6 144 84 70)(7 125 85 51)(8 142 86 68)(9 123 87 49)(10 140 88 66)(11 121 89 47)(12 138 90 64)(13 119 91 45)(14 136 92 62)(15 117 93 43)(16 134 94 60)(17 115 95 41)(18 132 96 58)(19 113 97 39)(20 130 98 56)(21 111 99 37)(22 128 100 54)(23 109 101 71)(24 126 102 52)(25 143 103 69)(26 124 104 50)(27 141 105 67)(28 122 106 48)(29 139 107 65)(30 120 108 46)(31 137 73 63)(32 118 74 44)(33 135 75 61)(34 116 76 42)(35 133 77 59)(36 114 78 40)
(1 48 19 66)(2 47 20 65)(3 46 21 64)(4 45 22 63)(5 44 23 62)(6 43 24 61)(7 42 25 60)(8 41 26 59)(9 40 27 58)(10 39 28 57)(11 38 29 56)(12 37 30 55)(13 72 31 54)(14 71 32 53)(15 70 33 52)(16 69 34 51)(17 68 35 50)(18 67 36 49)(73 128 91 110)(74 127 92 109)(75 126 93 144)(76 125 94 143)(77 124 95 142)(78 123 96 141)(79 122 97 140)(80 121 98 139)(81 120 99 138)(82 119 100 137)(83 118 101 136)(84 117 102 135)(85 116 103 134)(86 115 104 133)(87 114 105 132)(88 113 106 131)(89 112 107 130)(90 111 108 129)
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), (1,131,79,57)(2,112,80,38)(3,129,81,55)(4,110,82,72)(5,127,83,53)(6,144,84,70)(7,125,85,51)(8,142,86,68)(9,123,87,49)(10,140,88,66)(11,121,89,47)(12,138,90,64)(13,119,91,45)(14,136,92,62)(15,117,93,43)(16,134,94,60)(17,115,95,41)(18,132,96,58)(19,113,97,39)(20,130,98,56)(21,111,99,37)(22,128,100,54)(23,109,101,71)(24,126,102,52)(25,143,103,69)(26,124,104,50)(27,141,105,67)(28,122,106,48)(29,139,107,65)(30,120,108,46)(31,137,73,63)(32,118,74,44)(33,135,75,61)(34,116,76,42)(35,133,77,59)(36,114,78,40), (1,48,19,66)(2,47,20,65)(3,46,21,64)(4,45,22,63)(5,44,23,62)(6,43,24,61)(7,42,25,60)(8,41,26,59)(9,40,27,58)(10,39,28,57)(11,38,29,56)(12,37,30,55)(13,72,31,54)(14,71,32,53)(15,70,33,52)(16,69,34,51)(17,68,35,50)(18,67,36,49)(73,128,91,110)(74,127,92,109)(75,126,93,144)(76,125,94,143)(77,124,95,142)(78,123,96,141)(79,122,97,140)(80,121,98,139)(81,120,99,138)(82,119,100,137)(83,118,101,136)(84,117,102,135)(85,116,103,134)(86,115,104,133)(87,114,105,132)(88,113,106,131)(89,112,107,130)(90,111,108,129)>;
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), (1,131,79,57)(2,112,80,38)(3,129,81,55)(4,110,82,72)(5,127,83,53)(6,144,84,70)(7,125,85,51)(8,142,86,68)(9,123,87,49)(10,140,88,66)(11,121,89,47)(12,138,90,64)(13,119,91,45)(14,136,92,62)(15,117,93,43)(16,134,94,60)(17,115,95,41)(18,132,96,58)(19,113,97,39)(20,130,98,56)(21,111,99,37)(22,128,100,54)(23,109,101,71)(24,126,102,52)(25,143,103,69)(26,124,104,50)(27,141,105,67)(28,122,106,48)(29,139,107,65)(30,120,108,46)(31,137,73,63)(32,118,74,44)(33,135,75,61)(34,116,76,42)(35,133,77,59)(36,114,78,40), (1,48,19,66)(2,47,20,65)(3,46,21,64)(4,45,22,63)(5,44,23,62)(6,43,24,61)(7,42,25,60)(8,41,26,59)(9,40,27,58)(10,39,28,57)(11,38,29,56)(12,37,30,55)(13,72,31,54)(14,71,32,53)(15,70,33,52)(16,69,34,51)(17,68,35,50)(18,67,36,49)(73,128,91,110)(74,127,92,109)(75,126,93,144)(76,125,94,143)(77,124,95,142)(78,123,96,141)(79,122,97,140)(80,121,98,139)(81,120,99,138)(82,119,100,137)(83,118,101,136)(84,117,102,135)(85,116,103,134)(86,115,104,133)(87,114,105,132)(88,113,106,131)(89,112,107,130)(90,111,108,129) );
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)], [(1,131,79,57),(2,112,80,38),(3,129,81,55),(4,110,82,72),(5,127,83,53),(6,144,84,70),(7,125,85,51),(8,142,86,68),(9,123,87,49),(10,140,88,66),(11,121,89,47),(12,138,90,64),(13,119,91,45),(14,136,92,62),(15,117,93,43),(16,134,94,60),(17,115,95,41),(18,132,96,58),(19,113,97,39),(20,130,98,56),(21,111,99,37),(22,128,100,54),(23,109,101,71),(24,126,102,52),(25,143,103,69),(26,124,104,50),(27,141,105,67),(28,122,106,48),(29,139,107,65),(30,120,108,46),(31,137,73,63),(32,118,74,44),(33,135,75,61),(34,116,76,42),(35,133,77,59),(36,114,78,40)], [(1,48,19,66),(2,47,20,65),(3,46,21,64),(4,45,22,63),(5,44,23,62),(6,43,24,61),(7,42,25,60),(8,41,26,59),(9,40,27,58),(10,39,28,57),(11,38,29,56),(12,37,30,55),(13,72,31,54),(14,71,32,53),(15,70,33,52),(16,69,34,51),(17,68,35,50),(18,67,36,49),(73,128,91,110),(74,127,92,109),(75,126,93,144),(76,125,94,143),(77,124,95,142),(78,123,96,141),(79,122,97,140),(80,121,98,139),(81,120,99,138),(82,119,100,137),(83,118,101,136),(84,117,102,135),(85,116,103,134),(86,115,104,133),(87,114,105,132),(88,113,106,131),(89,112,107,130),(90,111,108,129)]])
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