metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊D23, C46.9D4, C4.3D46, C23⋊3SD16, D92.2C2, C92.3C22, C23⋊C8⋊3C2, (Q8×C23)⋊1C2, C2.6(C23⋊D4), SmallGroup(368,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D23
G = < a,b,c,d | a4=c23=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊D23
►On 184 points
Generators in S184
(1 88 37 63)(2 89 38 64)(3 90 39 65)(4 91 40 66)(5 92 41 67)(6 70 42 68)(7 71 43 69)(8 72 44 47)(9 73 45 48)(10 74 46 49)(11 75 24 50)(12 76 25 51)(13 77 26 52)(14 78 27 53)(15 79 28 54)(16 80 29 55)(17 81 30 56)(18 82 31 57)(19 83 32 58)(20 84 33 59)(21 85 34 60)(22 86 35 61)(23 87 36 62)(93 161 118 183)(94 139 119 184)(95 140 120 162)(96 141 121 163)(97 142 122 164)(98 143 123 165)(99 144 124 166)(100 145 125 167)(101 146 126 168)(102 147 127 169)(103 148 128 170)(104 149 129 171)(105 150 130 172)(106 151 131 173)(107 152 132 174)(108 153 133 175)(109 154 134 176)(110 155 135 177)(111 156 136 178)(112 157 137 179)(113 158 138 180)(114 159 116 181)(115 160 117 182)
(1 128 37 103)(2 129 38 104)(3 130 39 105)(4 131 40 106)(5 132 41 107)(6 133 42 108)(7 134 43 109)(8 135 44 110)(9 136 45 111)(10 137 46 112)(11 138 24 113)(12 116 25 114)(13 117 26 115)(14 118 27 93)(15 119 28 94)(16 120 29 95)(17 121 30 96)(18 122 31 97)(19 123 32 98)(20 124 33 99)(21 125 34 100)(22 126 35 101)(23 127 36 102)(47 177 72 155)(48 178 73 156)(49 179 74 157)(50 180 75 158)(51 181 76 159)(52 182 77 160)(53 183 78 161)(54 184 79 139)(55 162 80 140)(56 163 81 141)(57 164 82 142)(58 165 83 143)(59 166 84 144)(60 167 85 145)(61 168 86 146)(62 169 87 147)(63 170 88 148)(64 171 89 149)(65 172 90 150)(66 173 91 151)(67 174 92 152)(68 175 70 153)(69 176 71 154)
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161)(162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 26)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(93 157)(94 156)(95 155)(96 154)(97 153)(98 152)(99 151)(100 150)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 143)(108 142)(109 141)(110 140)(111 139)(112 161)(113 160)(114 159)(115 158)(116 181)(117 180)(118 179)(119 178)(120 177)(121 176)(122 175)(123 174)(124 173)(125 172)(126 171)(127 170)(128 169)(129 168)(130 167)(131 166)(132 165)(133 164)(134 163)(135 162)(136 184)(137 183)(138 182)
G:=sub<Sym(184)| (1,88,37,63)(2,89,38,64)(3,90,39,65)(4,91,40,66)(5,92,41,67)(6,70,42,68)(7,71,43,69)(8,72,44,47)(9,73,45,48)(10,74,46,49)(11,75,24,50)(12,76,25,51)(13,77,26,52)(14,78,27,53)(15,79,28,54)(16,80,29,55)(17,81,30,56)(18,82,31,57)(19,83,32,58)(20,84,33,59)(21,85,34,60)(22,86,35,61)(23,87,36,62)(93,161,118,183)(94,139,119,184)(95,140,120,162)(96,141,121,163)(97,142,122,164)(98,143,123,165)(99,144,124,166)(100,145,125,167)(101,146,126,168)(102,147,127,169)(103,148,128,170)(104,149,129,171)(105,150,130,172)(106,151,131,173)(107,152,132,174)(108,153,133,175)(109,154,134,176)(110,155,135,177)(111,156,136,178)(112,157,137,179)(113,158,138,180)(114,159,116,181)(115,160,117,182), (1,128,37,103)(2,129,38,104)(3,130,39,105)(4,131,40,106)(5,132,41,107)(6,133,42,108)(7,134,43,109)(8,135,44,110)(9,136,45,111)(10,137,46,112)(11,138,24,113)(12,116,25,114)(13,117,26,115)(14,118,27,93)(15,119,28,94)(16,120,29,95)(17,121,30,96)(18,122,31,97)(19,123,32,98)(20,124,33,99)(21,125,34,100)(22,126,35,101)(23,127,36,102)(47,177,72,155)(48,178,73,156)(49,179,74,157)(50,180,75,158)(51,181,76,159)(52,182,77,160)(53,183,78,161)(54,184,79,139)(55,162,80,140)(56,163,81,141)(57,164,82,142)(58,165,83,143)(59,166,84,144)(60,167,85,145)(61,168,86,146)(62,169,87,147)(63,170,88,148)(64,171,89,149)(65,172,90,150)(66,173,91,151)(67,174,92,152)(68,175,70,153)(69,176,71,154), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,26)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,140)(111,139)(112,161)(113,160)(114,159)(115,158)(116,181)(117,180)(118,179)(119,178)(120,177)(121,176)(122,175)(123,174)(124,173)(125,172)(126,171)(127,170)(128,169)(129,168)(130,167)(131,166)(132,165)(133,164)(134,163)(135,162)(136,184)(137,183)(138,182)>;
G:=Group( (1,88,37,63)(2,89,38,64)(3,90,39,65)(4,91,40,66)(5,92,41,67)(6,70,42,68)(7,71,43,69)(8,72,44,47)(9,73,45,48)(10,74,46,49)(11,75,24,50)(12,76,25,51)(13,77,26,52)(14,78,27,53)(15,79,28,54)(16,80,29,55)(17,81,30,56)(18,82,31,57)(19,83,32,58)(20,84,33,59)(21,85,34,60)(22,86,35,61)(23,87,36,62)(93,161,118,183)(94,139,119,184)(95,140,120,162)(96,141,121,163)(97,142,122,164)(98,143,123,165)(99,144,124,166)(100,145,125,167)(101,146,126,168)(102,147,127,169)(103,148,128,170)(104,149,129,171)(105,150,130,172)(106,151,131,173)(107,152,132,174)(108,153,133,175)(109,154,134,176)(110,155,135,177)(111,156,136,178)(112,157,137,179)(113,158,138,180)(114,159,116,181)(115,160,117,182), (1,128,37,103)(2,129,38,104)(3,130,39,105)(4,131,40,106)(5,132,41,107)(6,133,42,108)(7,134,43,109)(8,135,44,110)(9,136,45,111)(10,137,46,112)(11,138,24,113)(12,116,25,114)(13,117,26,115)(14,118,27,93)(15,119,28,94)(16,120,29,95)(17,121,30,96)(18,122,31,97)(19,123,32,98)(20,124,33,99)(21,125,34,100)(22,126,35,101)(23,127,36,102)(47,177,72,155)(48,178,73,156)(49,179,74,157)(50,180,75,158)(51,181,76,159)(52,182,77,160)(53,183,78,161)(54,184,79,139)(55,162,80,140)(56,163,81,141)(57,164,82,142)(58,165,83,143)(59,166,84,144)(60,167,85,145)(61,168,86,146)(62,169,87,147)(63,170,88,148)(64,171,89,149)(65,172,90,150)(66,173,91,151)(67,174,92,152)(68,175,70,153)(69,176,71,154), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,26)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,140)(111,139)(112,161)(113,160)(114,159)(115,158)(116,181)(117,180)(118,179)(119,178)(120,177)(121,176)(122,175)(123,174)(124,173)(125,172)(126,171)(127,170)(128,169)(129,168)(130,167)(131,166)(132,165)(133,164)(134,163)(135,162)(136,184)(137,183)(138,182) );
G=PermutationGroup([[(1,88,37,63),(2,89,38,64),(3,90,39,65),(4,91,40,66),(5,92,41,67),(6,70,42,68),(7,71,43,69),(8,72,44,47),(9,73,45,48),(10,74,46,49),(11,75,24,50),(12,76,25,51),(13,77,26,52),(14,78,27,53),(15,79,28,54),(16,80,29,55),(17,81,30,56),(18,82,31,57),(19,83,32,58),(20,84,33,59),(21,85,34,60),(22,86,35,61),(23,87,36,62),(93,161,118,183),(94,139,119,184),(95,140,120,162),(96,141,121,163),(97,142,122,164),(98,143,123,165),(99,144,124,166),(100,145,125,167),(101,146,126,168),(102,147,127,169),(103,148,128,170),(104,149,129,171),(105,150,130,172),(106,151,131,173),(107,152,132,174),(108,153,133,175),(109,154,134,176),(110,155,135,177),(111,156,136,178),(112,157,137,179),(113,158,138,180),(114,159,116,181),(115,160,117,182)], [(1,128,37,103),(2,129,38,104),(3,130,39,105),(4,131,40,106),(5,132,41,107),(6,133,42,108),(7,134,43,109),(8,135,44,110),(9,136,45,111),(10,137,46,112),(11,138,24,113),(12,116,25,114),(13,117,26,115),(14,118,27,93),(15,119,28,94),(16,120,29,95),(17,121,30,96),(18,122,31,97),(19,123,32,98),(20,124,33,99),(21,125,34,100),(22,126,35,101),(23,127,36,102),(47,177,72,155),(48,178,73,156),(49,179,74,157),(50,180,75,158),(51,181,76,159),(52,182,77,160),(53,183,78,161),(54,184,79,139),(55,162,80,140),(56,163,81,141),(57,164,82,142),(58,165,83,143),(59,166,84,144),(60,167,85,145),(61,168,86,146),(62,169,87,147),(63,170,88,148),(64,171,89,149),(65,172,90,150),(66,173,91,151),(67,174,92,152),(68,175,70,153),(69,176,71,154)], [(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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161),(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,26),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(47,80),(48,79),(49,78),(50,77),(51,76),(52,75),(53,74),(54,73),(55,72),(56,71),(57,70),(58,92),(59,91),(60,90),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(93,157),(94,156),(95,155),(96,154),(97,153),(98,152),(99,151),(100,150),(101,149),(102,148),(103,147),(104,146),(105,145),(106,144),(107,143),(108,142),(109,141),(110,140),(111,139),(112,161),(113,160),(114,159),(115,158),(116,181),(117,180),(118,179),(119,178),(120,177),(121,176),(122,175),(123,174),(124,173),(125,172),(126,171),(127,170),(128,169),(129,168),(130,167),(131,166),(132,165),(133,164),(134,163),(135,162),(136,184),(137,183),(138,182)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92AG |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 92 | 2 | 4 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D23 | D46 | C23⋊D4 | Q8⋊D23 |
| kernel | Q8⋊D23 | C23⋊C8 | D92 | Q8×C23 | C46 | C23 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 11 | 11 | 22 | 11 |
Matrix representation of Q8⋊D23 ►in GL4(𝔽1289) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1288 | 841 |
| 0 | 0 | 961 | 1 |
| 1288 | 0 | 0 | 0 |
| 0 | 1288 | 0 | 0 |
| 0 | 0 | 390 | 997 |
| 0 | 0 | 799 | 899 |
| 1206 | 1 | 0 | 0 |
| 494 | 491 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 1107 | 0 | 0 |
| 695 | 1276 | 0 | 0 |
| 0 | 0 | 1 | 448 |
| 0 | 0 | 0 | 1288 |
G:=sub<GL(4,GF(1289))| [1,0,0,0,0,1,0,0,0,0,1288,961,0,0,841,1],[1288,0,0,0,0,1288,0,0,0,0,390,799,0,0,997,899],[1206,494,0,0,1,491,0,0,0,0,1,0,0,0,0,1],[13,695,0,0,1107,1276,0,0,0,0,1,0,0,0,448,1288] >;
Q8⋊D23 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{23} % in TeX
G:=Group("Q8:D23"); // GroupNames label
G:=SmallGroup(368,16);
// by ID
G=gap.SmallGroup(368,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,61,46,182,97,42,8804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^23=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
Export