metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.D23, C46.11D4, C22⋊Dic23, C22.7D46, (C2×C46)⋊2C4, C46.9(C2×C4), C23⋊2(C22⋊C4), (C2×Dic23)⋊2C2, C2.3(C23⋊D4), (C22×C46).2C2, (C2×C46).7C22, C2.5(C2×Dic23), SmallGroup(368,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D23
G = < a,b,c,d,e | a2=b2=c2=d23=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Smallest permutation representation of C23.D23
►On 184 points
Generators in S184
(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)(139 162)(140 163)(141 164)(142 165)(143 166)(144 167)(145 168)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)(157 180)(158 181)(159 182)(160 183)(161 184)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 47)(20 48)(21 49)(22 50)(23 51)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 161)(116 162)(117 163)(118 164)(119 165)(120 166)(121 167)(122 168)(123 169)(124 170)(125 171)(126 172)(127 173)(128 174)(129 175)(130 176)(131 177)(132 178)(133 179)(134 180)(135 181)(136 182)(137 183)(138 184)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 24)(21 25)(22 26)(23 27)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 70)(65 71)(66 72)(67 73)(68 74)(69 75)(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)(139 162)(140 163)(141 164)(142 165)(143 166)(144 167)(145 168)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)(157 180)(158 181)(159 182)(160 183)(161 184)
(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)
(1 161 52 115)(2 160 53 114)(3 159 54 113)(4 158 55 112)(5 157 56 111)(6 156 57 110)(7 155 58 109)(8 154 59 108)(9 153 60 107)(10 152 61 106)(11 151 62 105)(12 150 63 104)(13 149 64 103)(14 148 65 102)(15 147 66 101)(16 146 67 100)(17 145 68 99)(18 144 69 98)(19 143 47 97)(20 142 48 96)(21 141 49 95)(22 140 50 94)(23 139 51 93)(24 165 77 119)(25 164 78 118)(26 163 79 117)(27 162 80 116)(28 184 81 138)(29 183 82 137)(30 182 83 136)(31 181 84 135)(32 180 85 134)(33 179 86 133)(34 178 87 132)(35 177 88 131)(36 176 89 130)(37 175 90 129)(38 174 91 128)(39 173 92 127)(40 172 70 126)(41 171 71 125)(42 170 72 124)(43 169 73 123)(44 168 74 122)(45 167 75 121)(46 166 76 120)
G:=sub<Sym(184)| (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)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,181)(159,182)(160,183)(161,184), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,47)(20,48)(21,49)(22,50)(23,51)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,24)(21,25)(22,26)(23,27)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,70)(65,71)(66,72)(67,73)(68,74)(69,75)(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)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,181)(159,182)(160,183)(161,184), (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), (1,161,52,115)(2,160,53,114)(3,159,54,113)(4,158,55,112)(5,157,56,111)(6,156,57,110)(7,155,58,109)(8,154,59,108)(9,153,60,107)(10,152,61,106)(11,151,62,105)(12,150,63,104)(13,149,64,103)(14,148,65,102)(15,147,66,101)(16,146,67,100)(17,145,68,99)(18,144,69,98)(19,143,47,97)(20,142,48,96)(21,141,49,95)(22,140,50,94)(23,139,51,93)(24,165,77,119)(25,164,78,118)(26,163,79,117)(27,162,80,116)(28,184,81,138)(29,183,82,137)(30,182,83,136)(31,181,84,135)(32,180,85,134)(33,179,86,133)(34,178,87,132)(35,177,88,131)(36,176,89,130)(37,175,90,129)(38,174,91,128)(39,173,92,127)(40,172,70,126)(41,171,71,125)(42,170,72,124)(43,169,73,123)(44,168,74,122)(45,167,75,121)(46,166,76,120)>;
G:=Group( (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)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,181)(159,182)(160,183)(161,184), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,47)(20,48)(21,49)(22,50)(23,51)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)(135,181)(136,182)(137,183)(138,184), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,24)(21,25)(22,26)(23,27)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,70)(65,71)(66,72)(67,73)(68,74)(69,75)(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)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,181)(159,182)(160,183)(161,184), (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), (1,161,52,115)(2,160,53,114)(3,159,54,113)(4,158,55,112)(5,157,56,111)(6,156,57,110)(7,155,58,109)(8,154,59,108)(9,153,60,107)(10,152,61,106)(11,151,62,105)(12,150,63,104)(13,149,64,103)(14,148,65,102)(15,147,66,101)(16,146,67,100)(17,145,68,99)(18,144,69,98)(19,143,47,97)(20,142,48,96)(21,141,49,95)(22,140,50,94)(23,139,51,93)(24,165,77,119)(25,164,78,118)(26,163,79,117)(27,162,80,116)(28,184,81,138)(29,183,82,137)(30,182,83,136)(31,181,84,135)(32,180,85,134)(33,179,86,133)(34,178,87,132)(35,177,88,131)(36,176,89,130)(37,175,90,129)(38,174,91,128)(39,173,92,127)(40,172,70,126)(41,171,71,125)(42,170,72,124)(43,169,73,123)(44,168,74,122)(45,167,75,121)(46,166,76,120) );
G=PermutationGroup([[(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),(139,162),(140,163),(141,164),(142,165),(143,166),(144,167),(145,168),(146,169),(147,170),(148,171),(149,172),(150,173),(151,174),(152,175),(153,176),(154,177),(155,178),(156,179),(157,180),(158,181),(159,182),(160,183),(161,184)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,47),(20,48),(21,49),(22,50),(23,51),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,161),(116,162),(117,163),(118,164),(119,165),(120,166),(121,167),(122,168),(123,169),(124,170),(125,171),(126,172),(127,173),(128,174),(129,175),(130,176),(131,177),(132,178),(133,179),(134,180),(135,181),(136,182),(137,183),(138,184)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,24),(21,25),(22,26),(23,27),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,85),(57,86),(58,87),(59,88),(60,89),(61,90),(62,91),(63,92),(64,70),(65,71),(66,72),(67,73),(68,74),(69,75),(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),(139,162),(140,163),(141,164),(142,165),(143,166),(144,167),(145,168),(146,169),(147,170),(148,171),(149,172),(150,173),(151,174),(152,175),(153,176),(154,177),(155,178),(156,179),(157,180),(158,181),(159,182),(160,183),(161,184)], [(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)], [(1,161,52,115),(2,160,53,114),(3,159,54,113),(4,158,55,112),(5,157,56,111),(6,156,57,110),(7,155,58,109),(8,154,59,108),(9,153,60,107),(10,152,61,106),(11,151,62,105),(12,150,63,104),(13,149,64,103),(14,148,65,102),(15,147,66,101),(16,146,67,100),(17,145,68,99),(18,144,69,98),(19,143,47,97),(20,142,48,96),(21,141,49,95),(22,140,50,94),(23,139,51,93),(24,165,77,119),(25,164,78,118),(26,163,79,117),(27,162,80,116),(28,184,81,138),(29,183,82,137),(30,182,83,136),(31,181,84,135),(32,180,85,134),(33,179,86,133),(34,178,87,132),(35,177,88,131),(36,176,89,130),(37,175,90,129),(38,174,91,128),(39,173,92,127),(40,172,70,126),(41,171,71,125),(42,170,72,124),(43,169,73,123),(44,168,74,122),(45,167,75,121),(46,166,76,120)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 23A | ··· | 23K | 46A | ··· | 46BY |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 46 | 46 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C4 | D4 | D23 | Dic23 | D46 | C23⋊D4 |
| kernel | C23.D23 | C2×Dic23 | C22×C46 | C2×C46 | C46 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 11 | 22 | 11 | 44 |
Matrix representation of C23.D23 ►in GL3(𝔽277) generated by
| 276 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 53 | 276 |
| 276 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 276 | 0 |
| 0 | 0 | 276 |
| 1 | 0 | 0 |
| 0 | 169 | 0 |
| 0 | 225 | 218 |
| 60 | 0 | 0 |
| 0 | 248 | 22 |
| 0 | 264 | 29 |
G:=sub<GL(3,GF(277))| [276,0,0,0,1,53,0,0,276],[276,0,0,0,1,0,0,0,1],[1,0,0,0,276,0,0,0,276],[1,0,0,0,169,225,0,0,218],[60,0,0,0,248,264,0,22,29] >;
C23.D23 in GAP, Magma, Sage, TeX
C_2^3.D_{23} % in TeX
G:=Group("C2^3.D23"); // GroupNames label
G:=SmallGroup(368,18);
// by ID
G=gap.SmallGroup(368,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,20,101,8804]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^23=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export