metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.15D18, C9⋊12- 1+4, C36.24C23, C18.10C24, D18.5C23, D36.13C22, Dic9.6C23, Dic18.13C22, (C2×Q8)⋊7D9, (Q8×D9)⋊4C2, (Q8×C18)⋊7C2, Q8⋊3D9⋊4C2, (C2×C4).22D18, (C3×Q8).61D6, (C6×Q8).22S3, D36⋊5C2⋊6C2, (C2×C12).103D6, C6.47(S3×C23), (C4×D9).5C22, C4.24(C22×D9), C2.11(C23×D9), C9⋊D4.2C22, (C2×C18).68C23, (C2×C36).51C22, C12.64(C22×S3), C3.(Q8.15D6), (Q8×C9).10C22, C22.7(C22×D9), (C2×C6).226(C22×S3), SmallGroup(288,361)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.15D18
G = < a,b,c,d | a4=c18=1, b2=d2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c-1 >
Subgroups: 824 in 219 conjugacy classes, 102 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C9, Dic3, C12, D6, C2×C6, C2×Q8, C2×Q8, C4○D4, D9, C18, C18, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, 2- 1+4, Dic9, C36, D18, C2×C18, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, Dic18, C4×D9, D36, C9⋊D4, C2×C36, Q8×C9, Q8.15D6, D36⋊5C2, Q8×D9, Q8⋊3D9, Q8×C18, Q8.15D18
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C22×S3, 2- 1+4, D18, S3×C23, C22×D9, Q8.15D6, C23×D9, Q8.15D18
►On 144 points
Generators in S144
(1 126 129 74)(2 109 130 75)(3 110 131 76)(4 111 132 77)(5 112 133 78)(6 113 134 79)(7 114 135 80)(8 115 136 81)(9 116 137 82)(10 117 138 83)(11 118 139 84)(12 119 140 85)(13 120 141 86)(14 121 142 87)(15 122 143 88)(16 123 144 89)(17 124 127 90)(18 125 128 73)(19 104 51 62)(20 105 52 63)(21 106 53 64)(22 107 54 65)(23 108 37 66)(24 91 38 67)(25 92 39 68)(26 93 40 69)(27 94 41 70)(28 95 42 71)(29 96 43 72)(30 97 44 55)(31 98 45 56)(32 99 46 57)(33 100 47 58)(34 101 48 59)(35 102 49 60)(36 103 50 61)
(1 105 129 63)(2 106 130 64)(3 107 131 65)(4 108 132 66)(5 91 133 67)(6 92 134 68)(7 93 135 69)(8 94 136 70)(9 95 137 71)(10 96 138 72)(11 97 139 55)(12 98 140 56)(13 99 141 57)(14 100 142 58)(15 101 143 59)(16 102 144 60)(17 103 127 61)(18 104 128 62)(19 73 51 125)(20 74 52 126)(21 75 53 109)(22 76 54 110)(23 77 37 111)(24 78 38 112)(25 79 39 113)(26 80 40 114)(27 81 41 115)(28 82 42 116)(29 83 43 117)(30 84 44 118)(31 85 45 119)(32 86 46 120)(33 87 47 121)(34 88 48 122)(35 89 49 123)(36 90 50 124)
(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 71 129 95)(2 94 130 70)(3 69 131 93)(4 92 132 68)(5 67 133 91)(6 108 134 66)(7 65 135 107)(8 106 136 64)(9 63 137 105)(10 104 138 62)(11 61 139 103)(12 102 140 60)(13 59 141 101)(14 100 142 58)(15 57 143 99)(16 98 144 56)(17 55 127 97)(18 96 128 72)(19 117 51 83)(20 82 52 116)(21 115 53 81)(22 80 54 114)(23 113 37 79)(24 78 38 112)(25 111 39 77)(26 76 40 110)(27 109 41 75)(28 74 42 126)(29 125 43 73)(30 90 44 124)(31 123 45 89)(32 88 46 122)(33 121 47 87)(34 86 48 120)(35 119 49 85)(36 84 50 118)
G:=sub<Sym(144)| (1,126,129,74)(2,109,130,75)(3,110,131,76)(4,111,132,77)(5,112,133,78)(6,113,134,79)(7,114,135,80)(8,115,136,81)(9,116,137,82)(10,117,138,83)(11,118,139,84)(12,119,140,85)(13,120,141,86)(14,121,142,87)(15,122,143,88)(16,123,144,89)(17,124,127,90)(18,125,128,73)(19,104,51,62)(20,105,52,63)(21,106,53,64)(22,107,54,65)(23,108,37,66)(24,91,38,67)(25,92,39,68)(26,93,40,69)(27,94,41,70)(28,95,42,71)(29,96,43,72)(30,97,44,55)(31,98,45,56)(32,99,46,57)(33,100,47,58)(34,101,48,59)(35,102,49,60)(36,103,50,61), (1,105,129,63)(2,106,130,64)(3,107,131,65)(4,108,132,66)(5,91,133,67)(6,92,134,68)(7,93,135,69)(8,94,136,70)(9,95,137,71)(10,96,138,72)(11,97,139,55)(12,98,140,56)(13,99,141,57)(14,100,142,58)(15,101,143,59)(16,102,144,60)(17,103,127,61)(18,104,128,62)(19,73,51,125)(20,74,52,126)(21,75,53,109)(22,76,54,110)(23,77,37,111)(24,78,38,112)(25,79,39,113)(26,80,40,114)(27,81,41,115)(28,82,42,116)(29,83,43,117)(30,84,44,118)(31,85,45,119)(32,86,46,120)(33,87,47,121)(34,88,48,122)(35,89,49,123)(36,90,50,124), (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), (1,71,129,95)(2,94,130,70)(3,69,131,93)(4,92,132,68)(5,67,133,91)(6,108,134,66)(7,65,135,107)(8,106,136,64)(9,63,137,105)(10,104,138,62)(11,61,139,103)(12,102,140,60)(13,59,141,101)(14,100,142,58)(15,57,143,99)(16,98,144,56)(17,55,127,97)(18,96,128,72)(19,117,51,83)(20,82,52,116)(21,115,53,81)(22,80,54,114)(23,113,37,79)(24,78,38,112)(25,111,39,77)(26,76,40,110)(27,109,41,75)(28,74,42,126)(29,125,43,73)(30,90,44,124)(31,123,45,89)(32,88,46,122)(33,121,47,87)(34,86,48,120)(35,119,49,85)(36,84,50,118)>;
G:=Group( (1,126,129,74)(2,109,130,75)(3,110,131,76)(4,111,132,77)(5,112,133,78)(6,113,134,79)(7,114,135,80)(8,115,136,81)(9,116,137,82)(10,117,138,83)(11,118,139,84)(12,119,140,85)(13,120,141,86)(14,121,142,87)(15,122,143,88)(16,123,144,89)(17,124,127,90)(18,125,128,73)(19,104,51,62)(20,105,52,63)(21,106,53,64)(22,107,54,65)(23,108,37,66)(24,91,38,67)(25,92,39,68)(26,93,40,69)(27,94,41,70)(28,95,42,71)(29,96,43,72)(30,97,44,55)(31,98,45,56)(32,99,46,57)(33,100,47,58)(34,101,48,59)(35,102,49,60)(36,103,50,61), (1,105,129,63)(2,106,130,64)(3,107,131,65)(4,108,132,66)(5,91,133,67)(6,92,134,68)(7,93,135,69)(8,94,136,70)(9,95,137,71)(10,96,138,72)(11,97,139,55)(12,98,140,56)(13,99,141,57)(14,100,142,58)(15,101,143,59)(16,102,144,60)(17,103,127,61)(18,104,128,62)(19,73,51,125)(20,74,52,126)(21,75,53,109)(22,76,54,110)(23,77,37,111)(24,78,38,112)(25,79,39,113)(26,80,40,114)(27,81,41,115)(28,82,42,116)(29,83,43,117)(30,84,44,118)(31,85,45,119)(32,86,46,120)(33,87,47,121)(34,88,48,122)(35,89,49,123)(36,90,50,124), (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), (1,71,129,95)(2,94,130,70)(3,69,131,93)(4,92,132,68)(5,67,133,91)(6,108,134,66)(7,65,135,107)(8,106,136,64)(9,63,137,105)(10,104,138,62)(11,61,139,103)(12,102,140,60)(13,59,141,101)(14,100,142,58)(15,57,143,99)(16,98,144,56)(17,55,127,97)(18,96,128,72)(19,117,51,83)(20,82,52,116)(21,115,53,81)(22,80,54,114)(23,113,37,79)(24,78,38,112)(25,111,39,77)(26,76,40,110)(27,109,41,75)(28,74,42,126)(29,125,43,73)(30,90,44,124)(31,123,45,89)(32,88,46,122)(33,121,47,87)(34,86,48,120)(35,119,49,85)(36,84,50,118) );
G=PermutationGroup([[(1,126,129,74),(2,109,130,75),(3,110,131,76),(4,111,132,77),(5,112,133,78),(6,113,134,79),(7,114,135,80),(8,115,136,81),(9,116,137,82),(10,117,138,83),(11,118,139,84),(12,119,140,85),(13,120,141,86),(14,121,142,87),(15,122,143,88),(16,123,144,89),(17,124,127,90),(18,125,128,73),(19,104,51,62),(20,105,52,63),(21,106,53,64),(22,107,54,65),(23,108,37,66),(24,91,38,67),(25,92,39,68),(26,93,40,69),(27,94,41,70),(28,95,42,71),(29,96,43,72),(30,97,44,55),(31,98,45,56),(32,99,46,57),(33,100,47,58),(34,101,48,59),(35,102,49,60),(36,103,50,61)], [(1,105,129,63),(2,106,130,64),(3,107,131,65),(4,108,132,66),(5,91,133,67),(6,92,134,68),(7,93,135,69),(8,94,136,70),(9,95,137,71),(10,96,138,72),(11,97,139,55),(12,98,140,56),(13,99,141,57),(14,100,142,58),(15,101,143,59),(16,102,144,60),(17,103,127,61),(18,104,128,62),(19,73,51,125),(20,74,52,126),(21,75,53,109),(22,76,54,110),(23,77,37,111),(24,78,38,112),(25,79,39,113),(26,80,40,114),(27,81,41,115),(28,82,42,116),(29,83,43,117),(30,84,44,118),(31,85,45,119),(32,86,46,120),(33,87,47,121),(34,88,48,122),(35,89,49,123),(36,90,50,124)], [(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)], [(1,71,129,95),(2,94,130,70),(3,69,131,93),(4,92,132,68),(5,67,133,91),(6,108,134,66),(7,65,135,107),(8,106,136,64),(9,63,137,105),(10,104,138,62),(11,61,139,103),(12,102,140,60),(13,59,141,101),(14,100,142,58),(15,57,143,99),(16,98,144,56),(17,55,127,97),(18,96,128,72),(19,117,51,83),(20,82,52,116),(21,115,53,81),(22,80,54,114),(23,113,37,79),(24,78,38,112),(25,111,39,77),(26,76,40,110),(27,109,41,75),(28,74,42,126),(29,125,43,73),(30,90,44,124),(31,123,45,89),(32,88,46,122),(33,121,47,87),(34,86,48,120),(35,119,49,85),(36,84,50,118)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D9 | D18 | D18 | 2- 1+4 | Q8.15D6 | Q8.15D18 |
| kernel | Q8.15D18 | D36⋊5C2 | Q8×D9 | Q8⋊3D9 | Q8×C18 | C6×Q8 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C9 | C3 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 1 | 3 | 4 | 3 | 9 | 12 | 1 | 2 | 6 |
Matrix representation of Q8.15D18 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 1 | 0 |
| 0 | 0 | 1 | 2 | 36 | 35 |
| 0 | 0 | 0 | 2 | 36 | 35 |
| 0 | 0 | 1 | 1 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 6 | 34 | 31 |
| 0 | 0 | 0 | 26 | 3 | 0 |
| 0 | 0 | 0 | 21 | 11 | 0 |
| 0 | 0 | 29 | 29 | 0 | 5 |
| 11 | 20 | 0 | 0 | 0 | 0 |
| 17 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 36 | 35 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 31 | 3 | 6 |
| 0 | 0 | 0 | 26 | 3 | 0 |
| 0 | 0 | 0 | 21 | 11 | 0 |
| 0 | 0 | 8 | 2 | 32 | 32 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,36,2,2,1,0,0,1,36,36,0,0,0,0,35,35,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,29,0,0,6,26,21,29,0,0,34,3,11,0,0,0,31,0,0,5],[11,17,0,0,0,0,20,31,0,0,0,0,0,0,0,1,2,0,0,0,1,0,2,0,0,0,0,0,36,0,0,0,0,0,35,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,8,0,0,31,26,21,2,0,0,3,3,11,32,0,0,6,0,0,32] >;
Q8.15D18 in GAP, Magma, Sage, TeX
Q_8._{15}D_{18} % in TeX
G:=Group("Q8.15D18"); // GroupNames label
G:=SmallGroup(288,361);
// by ID
G=gap.SmallGroup(288,361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^18=1,b^2=d^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations