metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D18⋊2Q8, C36.11D4, C4.13D36, C12.6D12, C4⋊C4⋊5D9, C2.7(Q8×D9), C4⋊Dic9⋊6C2, C18.8(C2×D4), C9⋊3(C22⋊Q8), C6.37(S3×Q8), D18⋊C4.2C2, (C2×C4).10D18, C6.37(C2×D12), C2.10(C2×D36), (C2×C12).10D6, C3.(C4.D12), C18.14(C2×Q8), (C2×Dic18)⋊7C2, C18.27(C4○D4), (C2×C18).38C23, (C2×C36).13C22, C2.13(D4⋊2D9), C6.83(D4⋊2S3), C22.52(C22×D9), (C2×Dic9).11C22, (C22×D9).22C22, (C9×C4⋊C4)⋊8C2, (C2×C4×D9).2C2, (C3×C4⋊C4).15S3, (C2×C6).195(C22×S3), SmallGroup(288,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18⋊2Q8
G = < a,b,c | a4=b36=1, c2=a2, bab-1=cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 500 in 111 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, C9, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, D9, C18, Dic6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊Q8, Dic9, C36, C36, D18, D18, C2×C18, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, Dic18, C4×D9, C2×Dic9, C2×Dic9, C2×C36, C2×C36, C22×D9, C4.D12, C4⋊Dic9, D18⋊C4, C9×C4⋊C4, C2×Dic18, C2×C4×D9, D18⋊2Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D9, D12, C22×S3, C22⋊Q8, D18, C2×D12, D4⋊2S3, S3×Q8, D36, C22×D9, C4.D12, C2×D36, D4⋊2D9, Q8×D9, D18⋊2Q8
►On 144 points
Generators in S144
(1 64 76 123)(2 124 77 65)(3 66 78 125)(4 126 79 67)(5 68 80 127)(6 128 81 69)(7 70 82 129)(8 130 83 71)(9 72 84 131)(10 132 85 37)(11 38 86 133)(12 134 87 39)(13 40 88 135)(14 136 89 41)(15 42 90 137)(16 138 91 43)(17 44 92 139)(18 140 93 45)(19 46 94 141)(20 142 95 47)(21 48 96 143)(22 144 97 49)(23 50 98 109)(24 110 99 51)(25 52 100 111)(26 112 101 53)(27 54 102 113)(28 114 103 55)(29 56 104 115)(30 116 105 57)(31 58 106 117)(32 118 107 59)(33 60 108 119)(34 120 73 61)(35 62 74 121)(36 122 75 63)
(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 75 76 36)(2 35 77 74)(3 73 78 34)(4 33 79 108)(5 107 80 32)(6 31 81 106)(7 105 82 30)(8 29 83 104)(9 103 84 28)(10 27 85 102)(11 101 86 26)(12 25 87 100)(13 99 88 24)(14 23 89 98)(15 97 90 22)(16 21 91 96)(17 95 92 20)(18 19 93 94)(37 54 132 113)(38 112 133 53)(39 52 134 111)(40 110 135 51)(41 50 136 109)(42 144 137 49)(43 48 138 143)(44 142 139 47)(45 46 140 141)(55 72 114 131)(56 130 115 71)(57 70 116 129)(58 128 117 69)(59 68 118 127)(60 126 119 67)(61 66 120 125)(62 124 121 65)(63 64 122 123)
G:=sub<Sym(144)| (1,64,76,123)(2,124,77,65)(3,66,78,125)(4,126,79,67)(5,68,80,127)(6,128,81,69)(7,70,82,129)(8,130,83,71)(9,72,84,131)(10,132,85,37)(11,38,86,133)(12,134,87,39)(13,40,88,135)(14,136,89,41)(15,42,90,137)(16,138,91,43)(17,44,92,139)(18,140,93,45)(19,46,94,141)(20,142,95,47)(21,48,96,143)(22,144,97,49)(23,50,98,109)(24,110,99,51)(25,52,100,111)(26,112,101,53)(27,54,102,113)(28,114,103,55)(29,56,104,115)(30,116,105,57)(31,58,106,117)(32,118,107,59)(33,60,108,119)(34,120,73,61)(35,62,74,121)(36,122,75,63), (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), (1,75,76,36)(2,35,77,74)(3,73,78,34)(4,33,79,108)(5,107,80,32)(6,31,81,106)(7,105,82,30)(8,29,83,104)(9,103,84,28)(10,27,85,102)(11,101,86,26)(12,25,87,100)(13,99,88,24)(14,23,89,98)(15,97,90,22)(16,21,91,96)(17,95,92,20)(18,19,93,94)(37,54,132,113)(38,112,133,53)(39,52,134,111)(40,110,135,51)(41,50,136,109)(42,144,137,49)(43,48,138,143)(44,142,139,47)(45,46,140,141)(55,72,114,131)(56,130,115,71)(57,70,116,129)(58,128,117,69)(59,68,118,127)(60,126,119,67)(61,66,120,125)(62,124,121,65)(63,64,122,123)>;
G:=Group( (1,64,76,123)(2,124,77,65)(3,66,78,125)(4,126,79,67)(5,68,80,127)(6,128,81,69)(7,70,82,129)(8,130,83,71)(9,72,84,131)(10,132,85,37)(11,38,86,133)(12,134,87,39)(13,40,88,135)(14,136,89,41)(15,42,90,137)(16,138,91,43)(17,44,92,139)(18,140,93,45)(19,46,94,141)(20,142,95,47)(21,48,96,143)(22,144,97,49)(23,50,98,109)(24,110,99,51)(25,52,100,111)(26,112,101,53)(27,54,102,113)(28,114,103,55)(29,56,104,115)(30,116,105,57)(31,58,106,117)(32,118,107,59)(33,60,108,119)(34,120,73,61)(35,62,74,121)(36,122,75,63), (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), (1,75,76,36)(2,35,77,74)(3,73,78,34)(4,33,79,108)(5,107,80,32)(6,31,81,106)(7,105,82,30)(8,29,83,104)(9,103,84,28)(10,27,85,102)(11,101,86,26)(12,25,87,100)(13,99,88,24)(14,23,89,98)(15,97,90,22)(16,21,91,96)(17,95,92,20)(18,19,93,94)(37,54,132,113)(38,112,133,53)(39,52,134,111)(40,110,135,51)(41,50,136,109)(42,144,137,49)(43,48,138,143)(44,142,139,47)(45,46,140,141)(55,72,114,131)(56,130,115,71)(57,70,116,129)(58,128,117,69)(59,68,118,127)(60,126,119,67)(61,66,120,125)(62,124,121,65)(63,64,122,123) );
G=PermutationGroup([[(1,64,76,123),(2,124,77,65),(3,66,78,125),(4,126,79,67),(5,68,80,127),(6,128,81,69),(7,70,82,129),(8,130,83,71),(9,72,84,131),(10,132,85,37),(11,38,86,133),(12,134,87,39),(13,40,88,135),(14,136,89,41),(15,42,90,137),(16,138,91,43),(17,44,92,139),(18,140,93,45),(19,46,94,141),(20,142,95,47),(21,48,96,143),(22,144,97,49),(23,50,98,109),(24,110,99,51),(25,52,100,111),(26,112,101,53),(27,54,102,113),(28,114,103,55),(29,56,104,115),(30,116,105,57),(31,58,106,117),(32,118,107,59),(33,60,108,119),(34,120,73,61),(35,62,74,121),(36,122,75,63)], [(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)], [(1,75,76,36),(2,35,77,74),(3,73,78,34),(4,33,79,108),(5,107,80,32),(6,31,81,106),(7,105,82,30),(8,29,83,104),(9,103,84,28),(10,27,85,102),(11,101,86,26),(12,25,87,100),(13,99,88,24),(14,23,89,98),(15,97,90,22),(16,21,91,96),(17,95,92,20),(18,19,93,94),(37,54,132,113),(38,112,133,53),(39,52,134,111),(40,110,135,51),(41,50,136,109),(42,144,137,49),(43,48,138,143),(44,142,139,47),(45,46,140,141),(55,72,114,131),(56,130,115,71),(57,70,116,129),(58,128,117,69),(59,68,118,127),(60,126,119,67),(61,66,120,125),(62,124,121,65),(63,64,122,123)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 9A | 9B | 9C | 12A | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | C4○D4 | D9 | D12 | D18 | D36 | D4⋊2S3 | S3×Q8 | D4⋊2D9 | Q8×D9 |
| kernel | D18⋊2Q8 | C4⋊Dic9 | D18⋊C4 | C9×C4⋊C4 | C2×Dic18 | C2×C4×D9 | C3×C4⋊C4 | C36 | D18 | C2×C12 | C18 | C4⋊C4 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 4 | 9 | 12 | 1 | 1 | 3 | 3 |
Matrix representation of D18⋊2Q8 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 3 | 31 |
| 29 | 12 | 0 | 0 |
| 25 | 4 | 0 | 0 |
| 0 | 0 | 27 | 3 |
| 0 | 0 | 4 | 10 |
| 25 | 8 | 0 | 0 |
| 33 | 12 | 0 | 0 |
| 0 | 0 | 27 | 3 |
| 0 | 0 | 28 | 10 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,6,3,0,0,0,31],[29,25,0,0,12,4,0,0,0,0,27,4,0,0,3,10],[25,33,0,0,8,12,0,0,0,0,27,28,0,0,3,10] >;
D18⋊2Q8 in GAP, Magma, Sage, TeX
D_{18}\rtimes_2Q_8 % in TeX
G:=Group("D18:2Q8"); // GroupNames label
G:=SmallGroup(288,107);
// by ID
G=gap.SmallGroup(288,107);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,142,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^4=b^36=1,c^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations