metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D23, C23⋊2D8, D92⋊2C2, C4.1D46, C46.7D4, C92.1C22, C23⋊C8⋊1C2, (D4×C23)⋊1C2, C2.4(C23⋊D4), SmallGroup(368,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D23
G = < a,b,c,d | a4=b2=c23=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D23
►On 184 points
Generators in S184
(1 75 25 66)(2 76 26 67)(3 77 27 68)(4 78 28 69)(5 79 29 47)(6 80 30 48)(7 81 31 49)(8 82 32 50)(9 83 33 51)(10 84 34 52)(11 85 35 53)(12 86 36 54)(13 87 37 55)(14 88 38 56)(15 89 39 57)(16 90 40 58)(17 91 41 59)(18 92 42 60)(19 70 43 61)(20 71 44 62)(21 72 45 63)(22 73 46 64)(23 74 24 65)(93 150 123 172)(94 151 124 173)(95 152 125 174)(96 153 126 175)(97 154 127 176)(98 155 128 177)(99 156 129 178)(100 157 130 179)(101 158 131 180)(102 159 132 181)(103 160 133 182)(104 161 134 183)(105 139 135 184)(106 140 136 162)(107 141 137 163)(108 142 138 164)(109 143 116 165)(110 144 117 166)(111 145 118 167)(112 146 119 168)(113 147 120 169)(114 148 121 170)(115 149 122 171)
(1 164)(2 165)(3 166)(4 167)(5 168)(6 169)(7 170)(8 171)(9 172)(10 173)(11 174)(12 175)(13 176)(14 177)(15 178)(16 179)(17 180)(18 181)(19 182)(20 183)(21 184)(22 162)(23 163)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(41 158)(42 159)(43 160)(44 161)(45 139)(46 140)(47 112)(48 113)(49 114)(50 115)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 116)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)
(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 25)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(47 70)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 84)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(93 156)(94 155)(95 154)(96 153)(97 152)(98 151)(99 150)(100 149)(101 148)(102 147)(103 146)(104 145)(105 144)(106 143)(107 142)(108 141)(109 140)(110 139)(111 161)(112 160)(113 159)(114 158)(115 157)(116 162)(117 184)(118 183)(119 182)(120 181)(121 180)(122 179)(123 178)(124 177)(125 176)(126 175)(127 174)(128 173)(129 172)(130 171)(131 170)(132 169)(133 168)(134 167)(135 166)(136 165)(137 164)(138 163)
G:=sub<Sym(184)| (1,75,25,66)(2,76,26,67)(3,77,27,68)(4,78,28,69)(5,79,29,47)(6,80,30,48)(7,81,31,49)(8,82,32,50)(9,83,33,51)(10,84,34,52)(11,85,35,53)(12,86,36,54)(13,87,37,55)(14,88,38,56)(15,89,39,57)(16,90,40,58)(17,91,41,59)(18,92,42,60)(19,70,43,61)(20,71,44,62)(21,72,45,63)(22,73,46,64)(23,74,24,65)(93,150,123,172)(94,151,124,173)(95,152,125,174)(96,153,126,175)(97,154,127,176)(98,155,128,177)(99,156,129,178)(100,157,130,179)(101,158,131,180)(102,159,132,181)(103,160,133,182)(104,161,134,183)(105,139,135,184)(106,140,136,162)(107,141,137,163)(108,142,138,164)(109,143,116,165)(110,144,117,166)(111,145,118,167)(112,146,119,168)(113,147,120,169)(114,148,121,170)(115,149,122,171), (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,162)(23,163)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,139)(46,140)(47,112)(48,113)(49,114)(50,115)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132), (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)(24,25)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(47,70)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,150)(100,149)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,140)(110,139)(111,161)(112,160)(113,159)(114,158)(115,157)(116,162)(117,184)(118,183)(119,182)(120,181)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,165)(137,164)(138,163)>;
G:=Group( (1,75,25,66)(2,76,26,67)(3,77,27,68)(4,78,28,69)(5,79,29,47)(6,80,30,48)(7,81,31,49)(8,82,32,50)(9,83,33,51)(10,84,34,52)(11,85,35,53)(12,86,36,54)(13,87,37,55)(14,88,38,56)(15,89,39,57)(16,90,40,58)(17,91,41,59)(18,92,42,60)(19,70,43,61)(20,71,44,62)(21,72,45,63)(22,73,46,64)(23,74,24,65)(93,150,123,172)(94,151,124,173)(95,152,125,174)(96,153,126,175)(97,154,127,176)(98,155,128,177)(99,156,129,178)(100,157,130,179)(101,158,131,180)(102,159,132,181)(103,160,133,182)(104,161,134,183)(105,139,135,184)(106,140,136,162)(107,141,137,163)(108,142,138,164)(109,143,116,165)(110,144,117,166)(111,145,118,167)(112,146,119,168)(113,147,120,169)(114,148,121,170)(115,149,122,171), (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,162)(23,163)(24,141)(25,142)(26,143)(27,144)(28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,139)(46,140)(47,112)(48,113)(49,114)(50,115)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132), (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)(24,25)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(47,70)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(93,156)(94,155)(95,154)(96,153)(97,152)(98,151)(99,150)(100,149)(101,148)(102,147)(103,146)(104,145)(105,144)(106,143)(107,142)(108,141)(109,140)(110,139)(111,161)(112,160)(113,159)(114,158)(115,157)(116,162)(117,184)(118,183)(119,182)(120,181)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,165)(137,164)(138,163) );
G=PermutationGroup([[(1,75,25,66),(2,76,26,67),(3,77,27,68),(4,78,28,69),(5,79,29,47),(6,80,30,48),(7,81,31,49),(8,82,32,50),(9,83,33,51),(10,84,34,52),(11,85,35,53),(12,86,36,54),(13,87,37,55),(14,88,38,56),(15,89,39,57),(16,90,40,58),(17,91,41,59),(18,92,42,60),(19,70,43,61),(20,71,44,62),(21,72,45,63),(22,73,46,64),(23,74,24,65),(93,150,123,172),(94,151,124,173),(95,152,125,174),(96,153,126,175),(97,154,127,176),(98,155,128,177),(99,156,129,178),(100,157,130,179),(101,158,131,180),(102,159,132,181),(103,160,133,182),(104,161,134,183),(105,139,135,184),(106,140,136,162),(107,141,137,163),(108,142,138,164),(109,143,116,165),(110,144,117,166),(111,145,118,167),(112,146,119,168),(113,147,120,169),(114,148,121,170),(115,149,122,171)], [(1,164),(2,165),(3,166),(4,167),(5,168),(6,169),(7,170),(8,171),(9,172),(10,173),(11,174),(12,175),(13,176),(14,177),(15,178),(16,179),(17,180),(18,181),(19,182),(20,183),(21,184),(22,162),(23,163),(24,141),(25,142),(26,143),(27,144),(28,145),(29,146),(30,147),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,157),(41,158),(42,159),(43,160),(44,161),(45,139),(46,140),(47,112),(48,113),(49,114),(50,115),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,133),(71,134),(72,135),(73,136),(74,137),(75,138),(76,116),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132)], [(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),(24,25),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(47,70),(48,92),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,84),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(93,156),(94,155),(95,154),(96,153),(97,152),(98,151),(99,150),(100,149),(101,148),(102,147),(103,146),(104,145),(105,144),(106,143),(107,142),(108,141),(109,140),(110,139),(111,161),(112,160),(113,159),(114,158),(115,157),(116,162),(117,184),(118,183),(119,182),(120,181),(121,180),(122,179),(123,178),(124,177),(125,176),(126,175),(127,174),(128,173),(129,172),(130,171),(131,170),(132,169),(133,168),(134,167),(135,166),(136,165),(137,164),(138,163)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 23A | ··· | 23K | 46A | ··· | 46K | 46L | ··· | 46AG | 92A | ··· | 92K |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 4 | 92 | 2 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D8 | D23 | D46 | C23⋊D4 | D4⋊D23 |
| kernel | D4⋊D23 | C23⋊C8 | D92 | D4×C23 | C46 | C23 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 11 | 11 | 22 | 11 |
Matrix representation of D4⋊D23 ►in GL4(𝔽1289) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 41 |
| 0 | 0 | 1006 | 1288 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 1197 | 0 |
| 1035 | 1 | 0 | 0 |
| 1074 | 1097 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1097 | 1288 | 0 | 0 |
| 771 | 192 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1006 | 1288 |
G:=sub<GL(4,GF(1289))| [1,0,0,0,0,1,0,0,0,0,1,1006,0,0,41,1288],[1,0,0,0,0,1,0,0,0,0,0,1197,0,0,14,0],[1035,1074,0,0,1,1097,0,0,0,0,1,0,0,0,0,1],[1097,771,0,0,1288,192,0,0,0,0,1,1006,0,0,0,1288] >;
D4⋊D23 in GAP, Magma, Sage, TeX
D_4\rtimes D_{23} % in TeX
G:=Group("D4:D23"); // GroupNames label
G:=SmallGroup(368,14);
// by ID
G=gap.SmallGroup(368,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,61,182,97,42,8804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^23=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export