direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D23, C184⋊3C2, D46.2C4, C4.12D46, C92.12C22, Dic23.2C4, C23⋊C8⋊6C2, C23⋊1(C2×C8), C46.1(C2×C4), C2.1(C4×D23), (C4×D23).3C2, SmallGroup(368,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C8×D23 |
Generators and relations for C8×D23
G = < a,b,c | a8=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D23
►On 184 points
Generators in S184
(1 162 76 137 33 152 49 95)(2 163 77 138 34 153 50 96)(3 164 78 116 35 154 51 97)(4 165 79 117 36 155 52 98)(5 166 80 118 37 156 53 99)(6 167 81 119 38 157 54 100)(7 168 82 120 39 158 55 101)(8 169 83 121 40 159 56 102)(9 170 84 122 41 160 57 103)(10 171 85 123 42 161 58 104)(11 172 86 124 43 139 59 105)(12 173 87 125 44 140 60 106)(13 174 88 126 45 141 61 107)(14 175 89 127 46 142 62 108)(15 176 90 128 24 143 63 109)(16 177 91 129 25 144 64 110)(17 178 92 130 26 145 65 111)(18 179 70 131 27 146 66 112)(19 180 71 132 28 147 67 113)(20 181 72 133 29 148 68 114)(21 182 73 134 30 149 69 115)(22 183 74 135 31 150 47 93)(23 184 75 136 32 151 48 94)
(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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(93 138)(94 137)(95 136)(96 135)(97 134)(98 133)(99 132)(100 131)(101 130)(102 129)(103 128)(104 127)(105 126)(106 125)(107 124)(108 123)(109 122)(110 121)(111 120)(112 119)(113 118)(114 117)(115 116)(139 174)(140 173)(141 172)(142 171)(143 170)(144 169)(145 168)(146 167)(147 166)(148 165)(149 164)(150 163)(151 162)(152 184)(153 183)(154 182)(155 181)(156 180)(157 179)(158 178)(159 177)(160 176)(161 175)
G:=sub<Sym(184)| (1,162,76,137,33,152,49,95)(2,163,77,138,34,153,50,96)(3,164,78,116,35,154,51,97)(4,165,79,117,36,155,52,98)(5,166,80,118,37,156,53,99)(6,167,81,119,38,157,54,100)(7,168,82,120,39,158,55,101)(8,169,83,121,40,159,56,102)(9,170,84,122,41,160,57,103)(10,171,85,123,42,161,58,104)(11,172,86,124,43,139,59,105)(12,173,87,125,44,140,60,106)(13,174,88,126,45,141,61,107)(14,175,89,127,46,142,62,108)(15,176,90,128,24,143,63,109)(16,177,91,129,25,144,64,110)(17,178,92,130,26,145,65,111)(18,179,70,131,27,146,66,112)(19,180,71,132,28,147,67,113)(20,181,72,133,29,148,68,114)(21,182,73,134,30,149,69,115)(22,183,74,135,31,150,47,93)(23,184,75,136,32,151,48,94), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(139,174)(140,173)(141,172)(142,171)(143,170)(144,169)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,184)(153,183)(154,182)(155,181)(156,180)(157,179)(158,178)(159,177)(160,176)(161,175)>;
G:=Group( (1,162,76,137,33,152,49,95)(2,163,77,138,34,153,50,96)(3,164,78,116,35,154,51,97)(4,165,79,117,36,155,52,98)(5,166,80,118,37,156,53,99)(6,167,81,119,38,157,54,100)(7,168,82,120,39,158,55,101)(8,169,83,121,40,159,56,102)(9,170,84,122,41,160,57,103)(10,171,85,123,42,161,58,104)(11,172,86,124,43,139,59,105)(12,173,87,125,44,140,60,106)(13,174,88,126,45,141,61,107)(14,175,89,127,46,142,62,108)(15,176,90,128,24,143,63,109)(16,177,91,129,25,144,64,110)(17,178,92,130,26,145,65,111)(18,179,70,131,27,146,66,112)(19,180,71,132,28,147,67,113)(20,181,72,133,29,148,68,114)(21,182,73,134,30,149,69,115)(22,183,74,135,31,150,47,93)(23,184,75,136,32,151,48,94), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(139,174)(140,173)(141,172)(142,171)(143,170)(144,169)(145,168)(146,167)(147,166)(148,165)(149,164)(150,163)(151,162)(152,184)(153,183)(154,182)(155,181)(156,180)(157,179)(158,178)(159,177)(160,176)(161,175) );
G=PermutationGroup([[(1,162,76,137,33,152,49,95),(2,163,77,138,34,153,50,96),(3,164,78,116,35,154,51,97),(4,165,79,117,36,155,52,98),(5,166,80,118,37,156,53,99),(6,167,81,119,38,157,54,100),(7,168,82,120,39,158,55,101),(8,169,83,121,40,159,56,102),(9,170,84,122,41,160,57,103),(10,171,85,123,42,161,58,104),(11,172,86,124,43,139,59,105),(12,173,87,125,44,140,60,106),(13,174,88,126,45,141,61,107),(14,175,89,127,46,142,62,108),(15,176,90,128,24,143,63,109),(16,177,91,129,25,144,64,110),(17,178,92,130,26,145,65,111),(18,179,70,131,27,146,66,112),(19,180,71,132,28,147,67,113),(20,181,72,133,29,148,68,114),(21,182,73,134,30,149,69,115),(22,183,74,135,31,150,47,93),(23,184,75,136,32,151,48,94)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(47,77),(48,76),(49,75),(50,74),(51,73),(52,72),(53,71),(54,70),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(93,138),(94,137),(95,136),(96,135),(97,134),(98,133),(99,132),(100,131),(101,130),(102,129),(103,128),(104,127),(105,126),(106,125),(107,124),(108,123),(109,122),(110,121),(111,120),(112,119),(113,118),(114,117),(115,116),(139,174),(140,173),(141,172),(142,171),(143,170),(144,169),(145,168),(146,167),(147,166),(148,165),(149,164),(150,163),(151,162),(152,184),(153,183),(154,182),(155,181),(156,180),(157,179),(158,178),(159,177),(160,176),(161,175)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V | 184A | ··· | 184AR |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 23 | 23 | 1 | 1 | 23 | 23 | 1 | 1 | 1 | 1 | 23 | 23 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D23 | D46 | C4×D23 | C8×D23 |
| kernel | C8×D23 | C23⋊C8 | C184 | C4×D23 | Dic23 | D46 | D23 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 11 | 11 | 22 | 44 |
Matrix representation of C8×D23 ►in GL3(𝔽1289) generated by
| 887 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 828 | 1 |
| 0 | 818 | 15 |
| 1 | 0 | 0 |
| 0 | 65 | 35 |
| 0 | 1021 | 1224 |
G:=sub<GL(3,GF(1289))| [887,0,0,0,1,0,0,0,1],[1,0,0,0,828,818,0,1,15],[1,0,0,0,65,1021,0,35,1224] >;
C8×D23 in GAP, Magma, Sage, TeX
C_8\times D_{23} % in TeX
G:=Group("C8xD23"); // GroupNames label
G:=SmallGroup(368,3);
// by ID
G=gap.SmallGroup(368,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,26,42,8804]);
// Polycyclic
G:=Group<a,b,c|a^8=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export