Aliases: Q8.C36, C24.2A4, C8○D4⋊C9, Q8⋊C9.C4, C6.6(C4×A4), C8.(C3.A4), C3.(C8.A4), C12.15(C2×A4), C4○D4.2C18, (C3×Q8).3C12, Q8.C18.3C2, (C3×C8○D4).C3, C4.5(C2×C3.A4), C2.3(C4×C3.A4), (C3×C4○D4).6C6, SmallGroup(288,77)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — Q8.C36 |
Generators and relations for Q8.C36
G = < a,b,c | a4=1, b2=c36=a2, bab-1=a-1, cac-1=b, cbc-1=ab >
Smallest permutation representation of Q8.C36
►On 144 points
Generators in S144
(1 87 37 123)(2 56 38 20)(3 107 39 143)(4 90 40 126)(5 59 41 23)(6 110 42 74)(7 93 43 129)(8 62 44 26)(9 113 45 77)(10 96 46 132)(11 65 47 29)(12 116 48 80)(13 99 49 135)(14 68 50 32)(15 119 51 83)(16 102 52 138)(17 71 53 35)(18 122 54 86)(19 105 55 141)(21 125 57 89)(22 108 58 144)(24 128 60 92)(25 111 61 75)(27 131 63 95)(28 114 64 78)(30 134 66 98)(31 117 67 81)(33 137 69 101)(34 120 70 84)(36 140 72 104)(73 91 109 127)(76 94 112 130)(79 97 115 133)(82 100 118 136)(85 103 121 139)(88 106 124 142)
(1 55 37 19)(2 106 38 142)(3 89 39 125)(4 58 40 22)(5 109 41 73)(6 92 42 128)(7 61 43 25)(8 112 44 76)(9 95 45 131)(10 64 46 28)(11 115 47 79)(12 98 48 134)(13 67 49 31)(14 118 50 82)(15 101 51 137)(16 70 52 34)(17 121 53 85)(18 104 54 140)(20 124 56 88)(21 107 57 143)(23 127 59 91)(24 110 60 74)(26 130 62 94)(27 113 63 77)(29 133 65 97)(30 116 66 80)(32 136 68 100)(33 119 69 83)(35 139 71 103)(36 122 72 86)(75 93 111 129)(78 96 114 132)(81 99 117 135)(84 102 120 138)(87 105 123 141)(90 108 126 144)
(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)
G:=sub<Sym(144)| (1,87,37,123)(2,56,38,20)(3,107,39,143)(4,90,40,126)(5,59,41,23)(6,110,42,74)(7,93,43,129)(8,62,44,26)(9,113,45,77)(10,96,46,132)(11,65,47,29)(12,116,48,80)(13,99,49,135)(14,68,50,32)(15,119,51,83)(16,102,52,138)(17,71,53,35)(18,122,54,86)(19,105,55,141)(21,125,57,89)(22,108,58,144)(24,128,60,92)(25,111,61,75)(27,131,63,95)(28,114,64,78)(30,134,66,98)(31,117,67,81)(33,137,69,101)(34,120,70,84)(36,140,72,104)(73,91,109,127)(76,94,112,130)(79,97,115,133)(82,100,118,136)(85,103,121,139)(88,106,124,142), (1,55,37,19)(2,106,38,142)(3,89,39,125)(4,58,40,22)(5,109,41,73)(6,92,42,128)(7,61,43,25)(8,112,44,76)(9,95,45,131)(10,64,46,28)(11,115,47,79)(12,98,48,134)(13,67,49,31)(14,118,50,82)(15,101,51,137)(16,70,52,34)(17,121,53,85)(18,104,54,140)(20,124,56,88)(21,107,57,143)(23,127,59,91)(24,110,60,74)(26,130,62,94)(27,113,63,77)(29,133,65,97)(30,116,66,80)(32,136,68,100)(33,119,69,83)(35,139,71,103)(36,122,72,86)(75,93,111,129)(78,96,114,132)(81,99,117,135)(84,102,120,138)(87,105,123,141)(90,108,126,144), (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)>;
G:=Group( (1,87,37,123)(2,56,38,20)(3,107,39,143)(4,90,40,126)(5,59,41,23)(6,110,42,74)(7,93,43,129)(8,62,44,26)(9,113,45,77)(10,96,46,132)(11,65,47,29)(12,116,48,80)(13,99,49,135)(14,68,50,32)(15,119,51,83)(16,102,52,138)(17,71,53,35)(18,122,54,86)(19,105,55,141)(21,125,57,89)(22,108,58,144)(24,128,60,92)(25,111,61,75)(27,131,63,95)(28,114,64,78)(30,134,66,98)(31,117,67,81)(33,137,69,101)(34,120,70,84)(36,140,72,104)(73,91,109,127)(76,94,112,130)(79,97,115,133)(82,100,118,136)(85,103,121,139)(88,106,124,142), (1,55,37,19)(2,106,38,142)(3,89,39,125)(4,58,40,22)(5,109,41,73)(6,92,42,128)(7,61,43,25)(8,112,44,76)(9,95,45,131)(10,64,46,28)(11,115,47,79)(12,98,48,134)(13,67,49,31)(14,118,50,82)(15,101,51,137)(16,70,52,34)(17,121,53,85)(18,104,54,140)(20,124,56,88)(21,107,57,143)(23,127,59,91)(24,110,60,74)(26,130,62,94)(27,113,63,77)(29,133,65,97)(30,116,66,80)(32,136,68,100)(33,119,69,83)(35,139,71,103)(36,122,72,86)(75,93,111,129)(78,96,114,132)(81,99,117,135)(84,102,120,138)(87,105,123,141)(90,108,126,144), (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) );
G=PermutationGroup([[(1,87,37,123),(2,56,38,20),(3,107,39,143),(4,90,40,126),(5,59,41,23),(6,110,42,74),(7,93,43,129),(8,62,44,26),(9,113,45,77),(10,96,46,132),(11,65,47,29),(12,116,48,80),(13,99,49,135),(14,68,50,32),(15,119,51,83),(16,102,52,138),(17,71,53,35),(18,122,54,86),(19,105,55,141),(21,125,57,89),(22,108,58,144),(24,128,60,92),(25,111,61,75),(27,131,63,95),(28,114,64,78),(30,134,66,98),(31,117,67,81),(33,137,69,101),(34,120,70,84),(36,140,72,104),(73,91,109,127),(76,94,112,130),(79,97,115,133),(82,100,118,136),(85,103,121,139),(88,106,124,142)], [(1,55,37,19),(2,106,38,142),(3,89,39,125),(4,58,40,22),(5,109,41,73),(6,92,42,128),(7,61,43,25),(8,112,44,76),(9,95,45,131),(10,64,46,28),(11,115,47,79),(12,98,48,134),(13,67,49,31),(14,118,50,82),(15,101,51,137),(16,70,52,34),(17,121,53,85),(18,104,54,140),(20,124,56,88),(21,107,57,143),(23,127,59,91),(24,110,60,74),(26,130,62,94),(27,113,63,77),(29,133,65,97),(30,116,66,80),(32,136,68,100),(33,119,69,83),(35,139,71,103),(36,122,72,86),(75,93,111,129),(78,96,114,132),(81,99,117,135),(84,102,120,138),(87,105,123,141),(90,108,126,144)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | ··· | 18F | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 6 | 1 | 1 | 1 | 1 | 6 | 1 | 1 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | |||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | C8.A4 | Q8.C36 | A4 | C2×A4 | C3.A4 | C4×A4 | C2×C3.A4 | C4×C3.A4 |
| kernel | Q8.C36 | Q8.C18 | C3×C8○D4 | Q8⋊C9 | C3×C4○D4 | C8○D4 | C3×Q8 | C4○D4 | Q8 | C3 | C1 | C24 | C12 | C8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 12 | 24 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of Q8.C36 ►in GL2(𝔽73) generated by
| 0 | 72 |
| 1 | 0 |
| 46 | 0 |
| 0 | 27 |
| 72 | 1 |
| 46 | 46 |
G:=sub<GL(2,GF(73))| [0,1,72,0],[46,0,0,27],[72,46,1,46] >;
Q8.C36 in GAP, Magma, Sage, TeX
Q_8.C_{36} % in TeX
G:=Group("Q8.C36"); // GroupNames label
G:=SmallGroup(288,77);
// by ID
G=gap.SmallGroup(288,77);
# by ID
G:=PCGroup([7,-2,-3,-2,-3,-2,2,-2,42,92,520,1271,172,2280,285,124]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^2=c^36=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,c*b*c^-1=a*b>;
// generators/relations
Export