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 [×5], C3, C4 [×6], C4 [×4], C22, C22 [×4], S3 [×4], C6, C6, C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], C9, Dic3 [×4], C12 [×6], D6 [×4], C2×C6, C2×Q8, C2×Q8 [×4], C4○D4 [×10], D9 [×4], C18, C18, Dic6 [×6], C4×S3 [×12], D12 [×6], C3⋊D4 [×4], C2×C12 [×3], C3×Q8 [×4], 2- 1+4, Dic9 [×4], C36 [×6], D18 [×4], C2×C18, C4○D12 [×6], S3×Q8 [×4], Q8⋊3S3 [×4], C6×Q8, Dic18 [×6], C4×D9 [×12], D36 [×6], C9⋊D4 [×4], C2×C36 [×3], Q8×C9 [×4], Q8.15D6, D36⋊5C2 [×6], Q8×D9 [×4], Q8⋊3D9 [×4], Q8×C18, Q8.15D18
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, D9, C22×S3 [×7], 2- 1+4, D18 [×7], S3×C23, C22×D9 [×7], Q8.15D6, C23×D9, Q8.15D18
►On 144 points
Generators in S144
(1 90 61 102)(2 73 62 103)(3 74 63 104)(4 75 64 105)(5 76 65 106)(6 77 66 107)(7 78 67 108)(8 79 68 91)(9 80 69 92)(10 81 70 93)(11 82 71 94)(12 83 72 95)(13 84 55 96)(14 85 56 97)(15 86 57 98)(16 87 58 99)(17 88 59 100)(18 89 60 101)(19 37 124 139)(20 38 125 140)(21 39 126 141)(22 40 109 142)(23 41 110 143)(24 42 111 144)(25 43 112 127)(26 44 113 128)(27 45 114 129)(28 46 115 130)(29 47 116 131)(30 48 117 132)(31 49 118 133)(32 50 119 134)(33 51 120 135)(34 52 121 136)(35 53 122 137)(36 54 123 138)
(1 110 61 23)(2 111 62 24)(3 112 63 25)(4 113 64 26)(5 114 65 27)(6 115 66 28)(7 116 67 29)(8 117 68 30)(9 118 69 31)(10 119 70 32)(11 120 71 33)(12 121 72 34)(13 122 55 35)(14 123 56 36)(15 124 57 19)(16 125 58 20)(17 126 59 21)(18 109 60 22)(37 98 139 86)(38 99 140 87)(39 100 141 88)(40 101 142 89)(41 102 143 90)(42 103 144 73)(43 104 127 74)(44 105 128 75)(45 106 129 76)(46 107 130 77)(47 108 131 78)(48 91 132 79)(49 92 133 80)(50 93 134 81)(51 94 135 82)(52 95 136 83)(53 96 137 84)(54 97 138 85)
(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 31 61 118)(2 117 62 30)(3 29 63 116)(4 115 64 28)(5 27 65 114)(6 113 66 26)(7 25 67 112)(8 111 68 24)(9 23 69 110)(10 109 70 22)(11 21 71 126)(12 125 72 20)(13 19 55 124)(14 123 56 36)(15 35 57 122)(16 121 58 34)(17 33 59 120)(18 119 60 32)(37 96 139 84)(38 83 140 95)(39 94 141 82)(40 81 142 93)(41 92 143 80)(42 79 144 91)(43 108 127 78)(44 77 128 107)(45 106 129 76)(46 75 130 105)(47 104 131 74)(48 73 132 103)(49 102 133 90)(50 89 134 101)(51 100 135 88)(52 87 136 99)(53 98 137 86)(54 85 138 97)
G:=sub<Sym(144)| (1,90,61,102)(2,73,62,103)(3,74,63,104)(4,75,64,105)(5,76,65,106)(6,77,66,107)(7,78,67,108)(8,79,68,91)(9,80,69,92)(10,81,70,93)(11,82,71,94)(12,83,72,95)(13,84,55,96)(14,85,56,97)(15,86,57,98)(16,87,58,99)(17,88,59,100)(18,89,60,101)(19,37,124,139)(20,38,125,140)(21,39,126,141)(22,40,109,142)(23,41,110,143)(24,42,111,144)(25,43,112,127)(26,44,113,128)(27,45,114,129)(28,46,115,130)(29,47,116,131)(30,48,117,132)(31,49,118,133)(32,50,119,134)(33,51,120,135)(34,52,121,136)(35,53,122,137)(36,54,123,138), (1,110,61,23)(2,111,62,24)(3,112,63,25)(4,113,64,26)(5,114,65,27)(6,115,66,28)(7,116,67,29)(8,117,68,30)(9,118,69,31)(10,119,70,32)(11,120,71,33)(12,121,72,34)(13,122,55,35)(14,123,56,36)(15,124,57,19)(16,125,58,20)(17,126,59,21)(18,109,60,22)(37,98,139,86)(38,99,140,87)(39,100,141,88)(40,101,142,89)(41,102,143,90)(42,103,144,73)(43,104,127,74)(44,105,128,75)(45,106,129,76)(46,107,130,77)(47,108,131,78)(48,91,132,79)(49,92,133,80)(50,93,134,81)(51,94,135,82)(52,95,136,83)(53,96,137,84)(54,97,138,85), (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), (1,31,61,118)(2,117,62,30)(3,29,63,116)(4,115,64,28)(5,27,65,114)(6,113,66,26)(7,25,67,112)(8,111,68,24)(9,23,69,110)(10,109,70,22)(11,21,71,126)(12,125,72,20)(13,19,55,124)(14,123,56,36)(15,35,57,122)(16,121,58,34)(17,33,59,120)(18,119,60,32)(37,96,139,84)(38,83,140,95)(39,94,141,82)(40,81,142,93)(41,92,143,80)(42,79,144,91)(43,108,127,78)(44,77,128,107)(45,106,129,76)(46,75,130,105)(47,104,131,74)(48,73,132,103)(49,102,133,90)(50,89,134,101)(51,100,135,88)(52,87,136,99)(53,98,137,86)(54,85,138,97)>;
G:=Group( (1,90,61,102)(2,73,62,103)(3,74,63,104)(4,75,64,105)(5,76,65,106)(6,77,66,107)(7,78,67,108)(8,79,68,91)(9,80,69,92)(10,81,70,93)(11,82,71,94)(12,83,72,95)(13,84,55,96)(14,85,56,97)(15,86,57,98)(16,87,58,99)(17,88,59,100)(18,89,60,101)(19,37,124,139)(20,38,125,140)(21,39,126,141)(22,40,109,142)(23,41,110,143)(24,42,111,144)(25,43,112,127)(26,44,113,128)(27,45,114,129)(28,46,115,130)(29,47,116,131)(30,48,117,132)(31,49,118,133)(32,50,119,134)(33,51,120,135)(34,52,121,136)(35,53,122,137)(36,54,123,138), (1,110,61,23)(2,111,62,24)(3,112,63,25)(4,113,64,26)(5,114,65,27)(6,115,66,28)(7,116,67,29)(8,117,68,30)(9,118,69,31)(10,119,70,32)(11,120,71,33)(12,121,72,34)(13,122,55,35)(14,123,56,36)(15,124,57,19)(16,125,58,20)(17,126,59,21)(18,109,60,22)(37,98,139,86)(38,99,140,87)(39,100,141,88)(40,101,142,89)(41,102,143,90)(42,103,144,73)(43,104,127,74)(44,105,128,75)(45,106,129,76)(46,107,130,77)(47,108,131,78)(48,91,132,79)(49,92,133,80)(50,93,134,81)(51,94,135,82)(52,95,136,83)(53,96,137,84)(54,97,138,85), (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), (1,31,61,118)(2,117,62,30)(3,29,63,116)(4,115,64,28)(5,27,65,114)(6,113,66,26)(7,25,67,112)(8,111,68,24)(9,23,69,110)(10,109,70,22)(11,21,71,126)(12,125,72,20)(13,19,55,124)(14,123,56,36)(15,35,57,122)(16,121,58,34)(17,33,59,120)(18,119,60,32)(37,96,139,84)(38,83,140,95)(39,94,141,82)(40,81,142,93)(41,92,143,80)(42,79,144,91)(43,108,127,78)(44,77,128,107)(45,106,129,76)(46,75,130,105)(47,104,131,74)(48,73,132,103)(49,102,133,90)(50,89,134,101)(51,100,135,88)(52,87,136,99)(53,98,137,86)(54,85,138,97) );
G=PermutationGroup([(1,90,61,102),(2,73,62,103),(3,74,63,104),(4,75,64,105),(5,76,65,106),(6,77,66,107),(7,78,67,108),(8,79,68,91),(9,80,69,92),(10,81,70,93),(11,82,71,94),(12,83,72,95),(13,84,55,96),(14,85,56,97),(15,86,57,98),(16,87,58,99),(17,88,59,100),(18,89,60,101),(19,37,124,139),(20,38,125,140),(21,39,126,141),(22,40,109,142),(23,41,110,143),(24,42,111,144),(25,43,112,127),(26,44,113,128),(27,45,114,129),(28,46,115,130),(29,47,116,131),(30,48,117,132),(31,49,118,133),(32,50,119,134),(33,51,120,135),(34,52,121,136),(35,53,122,137),(36,54,123,138)], [(1,110,61,23),(2,111,62,24),(3,112,63,25),(4,113,64,26),(5,114,65,27),(6,115,66,28),(7,116,67,29),(8,117,68,30),(9,118,69,31),(10,119,70,32),(11,120,71,33),(12,121,72,34),(13,122,55,35),(14,123,56,36),(15,124,57,19),(16,125,58,20),(17,126,59,21),(18,109,60,22),(37,98,139,86),(38,99,140,87),(39,100,141,88),(40,101,142,89),(41,102,143,90),(42,103,144,73),(43,104,127,74),(44,105,128,75),(45,106,129,76),(46,107,130,77),(47,108,131,78),(48,91,132,79),(49,92,133,80),(50,93,134,81),(51,94,135,82),(52,95,136,83),(53,96,137,84),(54,97,138,85)], [(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)], [(1,31,61,118),(2,117,62,30),(3,29,63,116),(4,115,64,28),(5,27,65,114),(6,113,66,26),(7,25,67,112),(8,111,68,24),(9,23,69,110),(10,109,70,22),(11,21,71,126),(12,125,72,20),(13,19,55,124),(14,123,56,36),(15,35,57,122),(16,121,58,34),(17,33,59,120),(18,119,60,32),(37,96,139,84),(38,83,140,95),(39,94,141,82),(40,81,142,93),(41,92,143,80),(42,79,144,91),(43,108,127,78),(44,77,128,107),(45,106,129,76),(46,75,130,105),(47,104,131,74),(48,73,132,103),(49,102,133,90),(50,89,134,101),(51,100,135,88),(52,87,136,99),(53,98,137,86),(54,85,138,97)])
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