metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D23, D46.C4, C184⋊4C2, C4.13D46, Dic23.C4, C23⋊1M4(2), C92.13C22, C23⋊C8⋊4C2, C46.2(C2×C4), C2.3(C4×D23), (C4×D23).2C2, SmallGroup(368,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D23
G = < a,b,c | a8=b23=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D23
►On 184 points
Generators in S184
(1 172 76 118 24 153 66 112)(2 173 77 119 25 154 67 113)(3 174 78 120 26 155 68 114)(4 175 79 121 27 156 69 115)(5 176 80 122 28 157 47 93)(6 177 81 123 29 158 48 94)(7 178 82 124 30 159 49 95)(8 179 83 125 31 160 50 96)(9 180 84 126 32 161 51 97)(10 181 85 127 33 139 52 98)(11 182 86 128 34 140 53 99)(12 183 87 129 35 141 54 100)(13 184 88 130 36 142 55 101)(14 162 89 131 37 143 56 102)(15 163 90 132 38 144 57 103)(16 164 91 133 39 145 58 104)(17 165 92 134 40 146 59 105)(18 166 70 135 41 147 60 106)(19 167 71 136 42 148 61 107)(20 168 72 137 43 149 62 108)(21 169 73 138 44 150 63 109)(22 170 74 116 45 151 64 110)(23 171 75 117 46 152 65 111)
(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 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(47 61)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(62 69)(63 68)(64 67)(65 66)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(82 92)(83 91)(84 90)(85 89)(86 88)(93 136)(94 135)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)(101 128)(102 127)(103 126)(104 125)(105 124)(106 123)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 138)(115 137)(139 162)(140 184)(141 183)(142 182)(143 181)(144 180)(145 179)(146 178)(147 177)(148 176)(149 175)(150 174)(151 173)(152 172)(153 171)(154 170)(155 169)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)
G:=sub<Sym(184)| (1,172,76,118,24,153,66,112)(2,173,77,119,25,154,67,113)(3,174,78,120,26,155,68,114)(4,175,79,121,27,156,69,115)(5,176,80,122,28,157,47,93)(6,177,81,123,29,158,48,94)(7,178,82,124,30,159,49,95)(8,179,83,125,31,160,50,96)(9,180,84,126,32,161,51,97)(10,181,85,127,33,139,52,98)(11,182,86,128,34,140,53,99)(12,183,87,129,35,141,54,100)(13,184,88,130,36,142,55,101)(14,162,89,131,37,143,56,102)(15,163,90,132,38,144,57,103)(16,164,91,133,39,145,58,104)(17,165,92,134,40,146,59,105)(18,166,70,135,41,147,60,106)(19,167,71,136,42,148,61,107)(20,168,72,137,43,149,62,108)(21,169,73,138,44,150,63,109)(22,170,74,116,45,151,64,110)(23,171,75,117,46,152,65,111), (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)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(62,69)(63,68)(64,67)(65,66)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(82,92)(83,91)(84,90)(85,89)(86,88)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,138)(115,137)(139,162)(140,184)(141,183)(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)>;
G:=Group( (1,172,76,118,24,153,66,112)(2,173,77,119,25,154,67,113)(3,174,78,120,26,155,68,114)(4,175,79,121,27,156,69,115)(5,176,80,122,28,157,47,93)(6,177,81,123,29,158,48,94)(7,178,82,124,30,159,49,95)(8,179,83,125,31,160,50,96)(9,180,84,126,32,161,51,97)(10,181,85,127,33,139,52,98)(11,182,86,128,34,140,53,99)(12,183,87,129,35,141,54,100)(13,184,88,130,36,142,55,101)(14,162,89,131,37,143,56,102)(15,163,90,132,38,144,57,103)(16,164,91,133,39,145,58,104)(17,165,92,134,40,146,59,105)(18,166,70,135,41,147,60,106)(19,167,71,136,42,148,61,107)(20,168,72,137,43,149,62,108)(21,169,73,138,44,150,63,109)(22,170,74,116,45,151,64,110)(23,171,75,117,46,152,65,111), (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)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(47,61)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(62,69)(63,68)(64,67)(65,66)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(82,92)(83,91)(84,90)(85,89)(86,88)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,138)(115,137)(139,162)(140,184)(141,183)(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163) );
G=PermutationGroup([[(1,172,76,118,24,153,66,112),(2,173,77,119,25,154,67,113),(3,174,78,120,26,155,68,114),(4,175,79,121,27,156,69,115),(5,176,80,122,28,157,47,93),(6,177,81,123,29,158,48,94),(7,178,82,124,30,159,49,95),(8,179,83,125,31,160,50,96),(9,180,84,126,32,161,51,97),(10,181,85,127,33,139,52,98),(11,182,86,128,34,140,53,99),(12,183,87,129,35,141,54,100),(13,184,88,130,36,142,55,101),(14,162,89,131,37,143,56,102),(15,163,90,132,38,144,57,103),(16,164,91,133,39,145,58,104),(17,165,92,134,40,146,59,105),(18,166,70,135,41,147,60,106),(19,167,71,136,42,148,61,107),(20,168,72,137,43,149,62,108),(21,169,73,138,44,150,63,109),(22,170,74,116,45,151,64,110),(23,171,75,117,46,152,65,111)], [(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),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(47,61),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(62,69),(63,68),(64,67),(65,66),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(82,92),(83,91),(84,90),(85,89),(86,88),(93,136),(94,135),(95,134),(96,133),(97,132),(98,131),(99,130),(100,129),(101,128),(102,127),(103,126),(104,125),(105,124),(106,123),(107,122),(108,121),(109,120),(110,119),(111,118),(112,117),(113,116),(114,138),(115,137),(139,162),(140,184),(141,183),(142,182),(143,181),(144,180),(145,179),(146,178),(147,177),(148,176),(149,175),(150,174),(151,173),(152,172),(153,171),(154,170),(155,169),(156,168),(157,167),(158,166),(159,165),(160,164),(161,163)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V | 184A | ··· | 184AR |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 46 | 1 | 1 | 46 | 2 | 2 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D23 | D46 | C4×D23 | C8⋊D23 |
| kernel | C8⋊D23 | C23⋊C8 | C184 | C4×D23 | Dic23 | D46 | C23 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 11 | 11 | 22 | 44 |
Matrix representation of C8⋊D23 ►in GL2(𝔽1289) generated by
| 235 | 614 |
| 675 | 1054 |
| 534 | 1 |
| 1288 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1289))| [235,675,614,1054],[534,1288,1,0],[0,1,1,0] >;
C8⋊D23 in GAP, Magma, Sage, TeX
C_8\rtimes D_{23} % in TeX
G:=Group("C8:D23"); // GroupNames label
G:=SmallGroup(368,4);
// by ID
G=gap.SmallGroup(368,4);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,101,26,42,8804]);
// Polycyclic
G:=Group<a,b,c|a^8=b^23=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
Export