metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D23, C46.8D4, C4.2D46, C23⋊2SD16, Dic46⋊2C2, C92.2C22, C23⋊C8⋊2C2, (D4×C23).1C2, C2.5(C23⋊D4), SmallGroup(368,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D23
G = < a,b,c,d | a4=b2=c23=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D23
►On 184 points
Generators in S184
(1 86 25 49)(2 87 26 50)(3 88 27 51)(4 89 28 52)(5 90 29 53)(6 91 30 54)(7 92 31 55)(8 70 32 56)(9 71 33 57)(10 72 34 58)(11 73 35 59)(12 74 36 60)(13 75 37 61)(14 76 38 62)(15 77 39 63)(16 78 40 64)(17 79 41 65)(18 80 42 66)(19 81 43 67)(20 82 44 68)(21 83 45 69)(22 84 46 47)(23 85 24 48)(93 139 116 162)(94 140 117 163)(95 141 118 164)(96 142 119 165)(97 143 120 166)(98 144 121 167)(99 145 122 168)(100 146 123 169)(101 147 124 170)(102 148 125 171)(103 149 126 172)(104 150 127 173)(105 151 128 174)(106 152 129 175)(107 153 130 176)(108 154 131 177)(109 155 132 178)(110 156 133 179)(111 157 134 180)(112 158 135 181)(113 159 136 182)(114 160 137 183)(115 161 138 184)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 47)(23 48)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)
(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)
(1 138 25 115)(2 137 26 114)(3 136 27 113)(4 135 28 112)(5 134 29 111)(6 133 30 110)(7 132 31 109)(8 131 32 108)(9 130 33 107)(10 129 34 106)(11 128 35 105)(12 127 36 104)(13 126 37 103)(14 125 38 102)(15 124 39 101)(16 123 40 100)(17 122 41 99)(18 121 42 98)(19 120 43 97)(20 119 44 96)(21 118 45 95)(22 117 46 94)(23 116 24 93)(47 163 84 140)(48 162 85 139)(49 184 86 161)(50 183 87 160)(51 182 88 159)(52 181 89 158)(53 180 90 157)(54 179 91 156)(55 178 92 155)(56 177 70 154)(57 176 71 153)(58 175 72 152)(59 174 73 151)(60 173 74 150)(61 172 75 149)(62 171 76 148)(63 170 77 147)(64 169 78 146)(65 168 79 145)(66 167 80 144)(67 166 81 143)(68 165 82 142)(69 164 83 141)
G:=sub<Sym(184)| (1,86,25,49)(2,87,26,50)(3,88,27,51)(4,89,28,52)(5,90,29,53)(6,91,30,54)(7,92,31,55)(8,70,32,56)(9,71,33,57)(10,72,34,58)(11,73,35,59)(12,74,36,60)(13,75,37,61)(14,76,38,62)(15,77,39,63)(16,78,40,64)(17,79,41,65)(18,80,42,66)(19,81,43,67)(20,82,44,68)(21,83,45,69)(22,84,46,47)(23,85,24,48)(93,139,116,162)(94,140,117,163)(95,141,118,164)(96,142,119,165)(97,143,120,166)(98,144,121,167)(99,145,122,168)(100,146,123,169)(101,147,124,170)(102,148,125,171)(103,149,126,172)(104,150,127,173)(105,151,128,174)(106,152,129,175)(107,153,130,176)(108,154,131,177)(109,155,132,178)(110,156,133,179)(111,157,134,180)(112,158,135,181)(113,159,136,182)(114,160,137,183)(115,161,138,184), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,47)(23,48)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138), (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), (1,138,25,115)(2,137,26,114)(3,136,27,113)(4,135,28,112)(5,134,29,111)(6,133,30,110)(7,132,31,109)(8,131,32,108)(9,130,33,107)(10,129,34,106)(11,128,35,105)(12,127,36,104)(13,126,37,103)(14,125,38,102)(15,124,39,101)(16,123,40,100)(17,122,41,99)(18,121,42,98)(19,120,43,97)(20,119,44,96)(21,118,45,95)(22,117,46,94)(23,116,24,93)(47,163,84,140)(48,162,85,139)(49,184,86,161)(50,183,87,160)(51,182,88,159)(52,181,89,158)(53,180,90,157)(54,179,91,156)(55,178,92,155)(56,177,70,154)(57,176,71,153)(58,175,72,152)(59,174,73,151)(60,173,74,150)(61,172,75,149)(62,171,76,148)(63,170,77,147)(64,169,78,146)(65,168,79,145)(66,167,80,144)(67,166,81,143)(68,165,82,142)(69,164,83,141)>;
G:=Group( (1,86,25,49)(2,87,26,50)(3,88,27,51)(4,89,28,52)(5,90,29,53)(6,91,30,54)(7,92,31,55)(8,70,32,56)(9,71,33,57)(10,72,34,58)(11,73,35,59)(12,74,36,60)(13,75,37,61)(14,76,38,62)(15,77,39,63)(16,78,40,64)(17,79,41,65)(18,80,42,66)(19,81,43,67)(20,82,44,68)(21,83,45,69)(22,84,46,47)(23,85,24,48)(93,139,116,162)(94,140,117,163)(95,141,118,164)(96,142,119,165)(97,143,120,166)(98,144,121,167)(99,145,122,168)(100,146,123,169)(101,147,124,170)(102,148,125,171)(103,149,126,172)(104,150,127,173)(105,151,128,174)(106,152,129,175)(107,153,130,176)(108,154,131,177)(109,155,132,178)(110,156,133,179)(111,157,134,180)(112,158,135,181)(113,159,136,182)(114,160,137,183)(115,161,138,184), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,47)(23,48)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,133)(111,134)(112,135)(113,136)(114,137)(115,138), (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), (1,138,25,115)(2,137,26,114)(3,136,27,113)(4,135,28,112)(5,134,29,111)(6,133,30,110)(7,132,31,109)(8,131,32,108)(9,130,33,107)(10,129,34,106)(11,128,35,105)(12,127,36,104)(13,126,37,103)(14,125,38,102)(15,124,39,101)(16,123,40,100)(17,122,41,99)(18,121,42,98)(19,120,43,97)(20,119,44,96)(21,118,45,95)(22,117,46,94)(23,116,24,93)(47,163,84,140)(48,162,85,139)(49,184,86,161)(50,183,87,160)(51,182,88,159)(52,181,89,158)(53,180,90,157)(54,179,91,156)(55,178,92,155)(56,177,70,154)(57,176,71,153)(58,175,72,152)(59,174,73,151)(60,173,74,150)(61,172,75,149)(62,171,76,148)(63,170,77,147)(64,169,78,146)(65,168,79,145)(66,167,80,144)(67,166,81,143)(68,165,82,142)(69,164,83,141) );
G=PermutationGroup([[(1,86,25,49),(2,87,26,50),(3,88,27,51),(4,89,28,52),(5,90,29,53),(6,91,30,54),(7,92,31,55),(8,70,32,56),(9,71,33,57),(10,72,34,58),(11,73,35,59),(12,74,36,60),(13,75,37,61),(14,76,38,62),(15,77,39,63),(16,78,40,64),(17,79,41,65),(18,80,42,66),(19,81,43,67),(20,82,44,68),(21,83,45,69),(22,84,46,47),(23,85,24,48),(93,139,116,162),(94,140,117,163),(95,141,118,164),(96,142,119,165),(97,143,120,166),(98,144,121,167),(99,145,122,168),(100,146,123,169),(101,147,124,170),(102,148,125,171),(103,149,126,172),(104,150,127,173),(105,151,128,174),(106,152,129,175),(107,153,130,176),(108,154,131,177),(109,155,132,178),(110,156,133,179),(111,157,134,180),(112,158,135,181),(113,159,136,182),(114,160,137,183),(115,161,138,184)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,47),(23,48),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,133),(111,134),(112,135),(113,136),(114,137),(115,138)], [(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)], [(1,138,25,115),(2,137,26,114),(3,136,27,113),(4,135,28,112),(5,134,29,111),(6,133,30,110),(7,132,31,109),(8,131,32,108),(9,130,33,107),(10,129,34,106),(11,128,35,105),(12,127,36,104),(13,126,37,103),(14,125,38,102),(15,124,39,101),(16,123,40,100),(17,122,41,99),(18,121,42,98),(19,120,43,97),(20,119,44,96),(21,118,45,95),(22,117,46,94),(23,116,24,93),(47,163,84,140),(48,162,85,139),(49,184,86,161),(50,183,87,160),(51,182,88,159),(52,181,89,158),(53,180,90,157),(54,179,91,156),(55,178,92,155),(56,177,70,154),(57,176,71,153),(58,175,72,152),(59,174,73,151),(60,173,74,150),(61,172,75,149),(62,171,76,148),(63,170,77,147),(64,169,78,146),(65,168,79,145),(66,167,80,144),(67,166,81,143),(68,165,82,142),(69,164,83,141)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 23A | ··· | 23K | 46A | ··· | 46K | 46L | ··· | 46AG | 92A | ··· | 92K |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 4 | 2 | 92 | 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 | SD16 | D23 | D46 | C23⋊D4 | D4.D23 |
| kernel | D4.D23 | C23⋊C8 | Dic46 | 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
| 1288 | 846 | 0 | 0 |
| 838 | 1 | 0 | 0 |
| 0 | 0 | 1288 | 0 |
| 0 | 0 | 0 | 1288 |
| 1288 | 846 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1288 | 0 |
| 0 | 0 | 923 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 444 | 0 |
| 0 | 0 | 332 | 90 |
| 390 | 22 | 0 | 0 |
| 293 | 899 | 0 | 0 |
| 0 | 0 | 430 | 54 |
| 0 | 0 | 801 | 859 |
G:=sub<GL(4,GF(1289))| [1288,838,0,0,846,1,0,0,0,0,1288,0,0,0,0,1288],[1288,0,0,0,846,1,0,0,0,0,1288,923,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,444,332,0,0,0,90],[390,293,0,0,22,899,0,0,0,0,430,801,0,0,54,859] >;
D4.D23 in GAP, Magma, Sage, TeX
D_4.D_{23} % in TeX
G:=Group("D4.D23"); // GroupNames label
G:=SmallGroup(368,15);
// by ID
G=gap.SmallGroup(368,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,40,61,182,97,42,8804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^23=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export