metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D23, C4.5D46, Dic46⋊3C2, C46.6C23, C92.5C22, C22.1D46, D46.2C22, Dic23.4C22, (C4×D23)⋊2C2, (D4×C23)⋊3C2, C23⋊2(C4○D4), C23⋊D4⋊2C2, (C2×C46).C22, (C2×Dic23)⋊3C2, C2.7(C22×D23), SmallGroup(368,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D23
G = < a,b,c,d | a4=b2=c23=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Smallest permutation representation of D4⋊2D23
►On 184 points
Generators in S184
(1 127 25 115)(2 128 26 93)(3 129 27 94)(4 130 28 95)(5 131 29 96)(6 132 30 97)(7 133 31 98)(8 134 32 99)(9 135 33 100)(10 136 34 101)(11 137 35 102)(12 138 36 103)(13 116 37 104)(14 117 38 105)(15 118 39 106)(16 119 40 107)(17 120 41 108)(18 121 42 109)(19 122 43 110)(20 123 44 111)(21 124 45 112)(22 125 46 113)(23 126 24 114)(47 173 75 145)(48 174 76 146)(49 175 77 147)(50 176 78 148)(51 177 79 149)(52 178 80 150)(53 179 81 151)(54 180 82 152)(55 181 83 153)(56 182 84 154)(57 183 85 155)(58 184 86 156)(59 162 87 157)(60 163 88 158)(61 164 89 159)(62 165 90 160)(63 166 91 161)(64 167 92 139)(65 168 70 140)(66 169 71 141)(67 170 72 142)(68 171 73 143)(69 172 74 144)
(1 152)(2 153)(3 154)(4 155)(5 156)(6 157)(7 158)(8 159)(9 160)(10 161)(11 139)(12 140)(13 141)(14 142)(15 143)(16 144)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 179)(25 180)(26 181)(27 182)(28 183)(29 184)(30 162)(31 163)(32 164)(33 165)(34 166)(35 167)(36 168)(37 169)(38 170)(39 171)(40 172)(41 173)(42 174)(43 175)(44 176)(45 177)(46 178)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 138)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)
(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 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 92)(67 91)(68 90)(69 89)(93 113)(94 112)(95 111)(96 110)(97 109)(98 108)(99 107)(100 106)(101 105)(102 104)(114 115)(116 137)(117 136)(118 135)(119 134)(120 133)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(139 169)(140 168)(141 167)(142 166)(143 165)(144 164)(145 163)(146 162)(147 184)(148 183)(149 182)(150 181)(151 180)(152 179)(153 178)(154 177)(155 176)(156 175)(157 174)(158 173)(159 172)(160 171)(161 170)
G:=sub<Sym(184)| (1,127,25,115)(2,128,26,93)(3,129,27,94)(4,130,28,95)(5,131,29,96)(6,132,30,97)(7,133,31,98)(8,134,32,99)(9,135,33,100)(10,136,34,101)(11,137,35,102)(12,138,36,103)(13,116,37,104)(14,117,38,105)(15,118,39,106)(16,119,40,107)(17,120,41,108)(18,121,42,109)(19,122,43,110)(20,123,44,111)(21,124,45,112)(22,125,46,113)(23,126,24,114)(47,173,75,145)(48,174,76,146)(49,175,77,147)(50,176,78,148)(51,177,79,149)(52,178,80,150)(53,179,81,151)(54,180,82,152)(55,181,83,153)(56,182,84,154)(57,183,85,155)(58,184,86,156)(59,162,87,157)(60,163,88,158)(61,164,89,159)(62,165,90,160)(63,166,91,161)(64,167,92,139)(65,168,70,140)(66,169,71,141)(67,170,72,142)(68,171,73,143)(69,172,74,144), (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,139)(12,140)(13,141)(14,142)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,179)(25,180)(26,181)(27,182)(28,183)(29,184)(30,162)(31,163)(32,164)(33,165)(34,166)(35,167)(36,168)(37,169)(38,170)(39,171)(40,172)(41,173)(42,174)(43,175)(44,176)(45,177)(46,178)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,138)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137), (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)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,92)(67,91)(68,90)(69,89)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(114,115)(116,137)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(139,169)(140,168)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,184)(148,183)(149,182)(150,181)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170)>;
G:=Group( (1,127,25,115)(2,128,26,93)(3,129,27,94)(4,130,28,95)(5,131,29,96)(6,132,30,97)(7,133,31,98)(8,134,32,99)(9,135,33,100)(10,136,34,101)(11,137,35,102)(12,138,36,103)(13,116,37,104)(14,117,38,105)(15,118,39,106)(16,119,40,107)(17,120,41,108)(18,121,42,109)(19,122,43,110)(20,123,44,111)(21,124,45,112)(22,125,46,113)(23,126,24,114)(47,173,75,145)(48,174,76,146)(49,175,77,147)(50,176,78,148)(51,177,79,149)(52,178,80,150)(53,179,81,151)(54,180,82,152)(55,181,83,153)(56,182,84,154)(57,183,85,155)(58,184,86,156)(59,162,87,157)(60,163,88,158)(61,164,89,159)(62,165,90,160)(63,166,91,161)(64,167,92,139)(65,168,70,140)(66,169,71,141)(67,170,72,142)(68,171,73,143)(69,172,74,144), (1,152)(2,153)(3,154)(4,155)(5,156)(6,157)(7,158)(8,159)(9,160)(10,161)(11,139)(12,140)(13,141)(14,142)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,179)(25,180)(26,181)(27,182)(28,183)(29,184)(30,162)(31,163)(32,164)(33,165)(34,166)(35,167)(36,168)(37,169)(38,170)(39,171)(40,172)(41,173)(42,174)(43,175)(44,176)(45,177)(46,178)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,93)(56,94)(57,95)(58,96)(59,97)(60,98)(61,99)(62,100)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,138)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,127)(83,128)(84,129)(85,130)(86,131)(87,132)(88,133)(89,134)(90,135)(91,136)(92,137), (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)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,92)(67,91)(68,90)(69,89)(93,113)(94,112)(95,111)(96,110)(97,109)(98,108)(99,107)(100,106)(101,105)(102,104)(114,115)(116,137)(117,136)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(139,169)(140,168)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,184)(148,183)(149,182)(150,181)(151,180)(152,179)(153,178)(154,177)(155,176)(156,175)(157,174)(158,173)(159,172)(160,171)(161,170) );
G=PermutationGroup([[(1,127,25,115),(2,128,26,93),(3,129,27,94),(4,130,28,95),(5,131,29,96),(6,132,30,97),(7,133,31,98),(8,134,32,99),(9,135,33,100),(10,136,34,101),(11,137,35,102),(12,138,36,103),(13,116,37,104),(14,117,38,105),(15,118,39,106),(16,119,40,107),(17,120,41,108),(18,121,42,109),(19,122,43,110),(20,123,44,111),(21,124,45,112),(22,125,46,113),(23,126,24,114),(47,173,75,145),(48,174,76,146),(49,175,77,147),(50,176,78,148),(51,177,79,149),(52,178,80,150),(53,179,81,151),(54,180,82,152),(55,181,83,153),(56,182,84,154),(57,183,85,155),(58,184,86,156),(59,162,87,157),(60,163,88,158),(61,164,89,159),(62,165,90,160),(63,166,91,161),(64,167,92,139),(65,168,70,140),(66,169,71,141),(67,170,72,142),(68,171,73,143),(69,172,74,144)], [(1,152),(2,153),(3,154),(4,155),(5,156),(6,157),(7,158),(8,159),(9,160),(10,161),(11,139),(12,140),(13,141),(14,142),(15,143),(16,144),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,179),(25,180),(26,181),(27,182),(28,183),(29,184),(30,162),(31,163),(32,164),(33,165),(34,166),(35,167),(36,168),(37,169),(38,170),(39,171),(40,172),(41,173),(42,174),(43,175),(44,176),(45,177),(46,178),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,93),(56,94),(57,95),(58,96),(59,97),(60,98),(61,99),(62,100),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,138),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,127),(83,128),(84,129),(85,130),(86,131),(87,132),(88,133),(89,134),(90,135),(91,136),(92,137)], [(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),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,92),(67,91),(68,90),(69,89),(93,113),(94,112),(95,111),(96,110),(97,109),(98,108),(99,107),(100,106),(101,105),(102,104),(114,115),(116,137),(117,136),(118,135),(119,134),(120,133),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(139,169),(140,168),(141,167),(142,166),(143,165),(144,164),(145,163),(146,162),(147,184),(148,183),(149,182),(150,181),(151,180),(152,179),(153,178),(154,177),(155,176),(156,175),(157,174),(158,173),(159,172),(160,171),(161,170)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 23A | ··· | 23K | 46A | ··· | 46K | 46L | ··· | 46AG | 92A | ··· | 92K |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 2 | 2 | 46 | 2 | 23 | 23 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D23 | D46 | D46 | D4⋊2D23 |
| kernel | D4⋊2D23 | Dic46 | C4×D23 | C2×Dic23 | C23⋊D4 | D4×C23 | C23 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 11 | 11 | 22 | 11 |
Matrix representation of D4⋊2D23 ►in GL4(𝔽277) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 217 | 0 |
| 0 | 0 | 65 | 60 |
| 276 | 0 | 0 | 0 |
| 0 | 276 | 0 | 0 |
| 0 | 0 | 89 | 143 |
| 0 | 0 | 179 | 188 |
| 62 | 1 | 0 | 0 |
| 185 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 132 | 227 | 0 | 0 |
| 77 | 145 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 22 | 276 |
G:=sub<GL(4,GF(277))| [1,0,0,0,0,1,0,0,0,0,217,65,0,0,0,60],[276,0,0,0,0,276,0,0,0,0,89,179,0,0,143,188],[62,185,0,0,1,3,0,0,0,0,1,0,0,0,0,1],[132,77,0,0,227,145,0,0,0,0,1,22,0,0,0,276] >;
D4⋊2D23 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_{23} % in TeX
G:=Group("D4:2D23"); // GroupNames label
G:=SmallGroup(368,32);
// by ID
G=gap.SmallGroup(368,32);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,46,182,97,8804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^23=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
Export