direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C23, C4.C46, C92.3C2, C46.7C22, C2.2(C2×C46), SmallGroup(184,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C23
G = < a,b,c | a23=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C23
►Regular action on 184 points
Generators in S184
(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 96 69 134)(2 97 47 135)(3 98 48 136)(4 99 49 137)(5 100 50 138)(6 101 51 116)(7 102 52 117)(8 103 53 118)(9 104 54 119)(10 105 55 120)(11 106 56 121)(12 107 57 122)(13 108 58 123)(14 109 59 124)(15 110 60 125)(16 111 61 126)(17 112 62 127)(18 113 63 128)(19 114 64 129)(20 115 65 130)(21 93 66 131)(22 94 67 132)(23 95 68 133)(24 156 85 178)(25 157 86 179)(26 158 87 180)(27 159 88 181)(28 160 89 182)(29 161 90 183)(30 139 91 184)(31 140 92 162)(32 141 70 163)(33 142 71 164)(34 143 72 165)(35 144 73 166)(36 145 74 167)(37 146 75 168)(38 147 76 169)(39 148 77 170)(40 149 78 171)(41 150 79 172)(42 151 80 173)(43 152 81 174)(44 153 82 175)(45 154 83 176)(46 155 84 177)
(1 80 69 42)(2 81 47 43)(3 82 48 44)(4 83 49 45)(5 84 50 46)(6 85 51 24)(7 86 52 25)(8 87 53 26)(9 88 54 27)(10 89 55 28)(11 90 56 29)(12 91 57 30)(13 92 58 31)(14 70 59 32)(15 71 60 33)(16 72 61 34)(17 73 62 35)(18 74 63 36)(19 75 64 37)(20 76 65 38)(21 77 66 39)(22 78 67 40)(23 79 68 41)(93 148 131 170)(94 149 132 171)(95 150 133 172)(96 151 134 173)(97 152 135 174)(98 153 136 175)(99 154 137 176)(100 155 138 177)(101 156 116 178)(102 157 117 179)(103 158 118 180)(104 159 119 181)(105 160 120 182)(106 161 121 183)(107 139 122 184)(108 140 123 162)(109 141 124 163)(110 142 125 164)(111 143 126 165)(112 144 127 166)(113 145 128 167)(114 146 129 168)(115 147 130 169)
G:=sub<Sym(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), (1,96,69,134)(2,97,47,135)(3,98,48,136)(4,99,49,137)(5,100,50,138)(6,101,51,116)(7,102,52,117)(8,103,53,118)(9,104,54,119)(10,105,55,120)(11,106,56,121)(12,107,57,122)(13,108,58,123)(14,109,59,124)(15,110,60,125)(16,111,61,126)(17,112,62,127)(18,113,63,128)(19,114,64,129)(20,115,65,130)(21,93,66,131)(22,94,67,132)(23,95,68,133)(24,156,85,178)(25,157,86,179)(26,158,87,180)(27,159,88,181)(28,160,89,182)(29,161,90,183)(30,139,91,184)(31,140,92,162)(32,141,70,163)(33,142,71,164)(34,143,72,165)(35,144,73,166)(36,145,74,167)(37,146,75,168)(38,147,76,169)(39,148,77,170)(40,149,78,171)(41,150,79,172)(42,151,80,173)(43,152,81,174)(44,153,82,175)(45,154,83,176)(46,155,84,177), (1,80,69,42)(2,81,47,43)(3,82,48,44)(4,83,49,45)(5,84,50,46)(6,85,51,24)(7,86,52,25)(8,87,53,26)(9,88,54,27)(10,89,55,28)(11,90,56,29)(12,91,57,30)(13,92,58,31)(14,70,59,32)(15,71,60,33)(16,72,61,34)(17,73,62,35)(18,74,63,36)(19,75,64,37)(20,76,65,38)(21,77,66,39)(22,78,67,40)(23,79,68,41)(93,148,131,170)(94,149,132,171)(95,150,133,172)(96,151,134,173)(97,152,135,174)(98,153,136,175)(99,154,137,176)(100,155,138,177)(101,156,116,178)(102,157,117,179)(103,158,118,180)(104,159,119,181)(105,160,120,182)(106,161,121,183)(107,139,122,184)(108,140,123,162)(109,141,124,163)(110,142,125,164)(111,143,126,165)(112,144,127,166)(113,145,128,167)(114,146,129,168)(115,147,130,169)>;
G:=Group( (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), (1,96,69,134)(2,97,47,135)(3,98,48,136)(4,99,49,137)(5,100,50,138)(6,101,51,116)(7,102,52,117)(8,103,53,118)(9,104,54,119)(10,105,55,120)(11,106,56,121)(12,107,57,122)(13,108,58,123)(14,109,59,124)(15,110,60,125)(16,111,61,126)(17,112,62,127)(18,113,63,128)(19,114,64,129)(20,115,65,130)(21,93,66,131)(22,94,67,132)(23,95,68,133)(24,156,85,178)(25,157,86,179)(26,158,87,180)(27,159,88,181)(28,160,89,182)(29,161,90,183)(30,139,91,184)(31,140,92,162)(32,141,70,163)(33,142,71,164)(34,143,72,165)(35,144,73,166)(36,145,74,167)(37,146,75,168)(38,147,76,169)(39,148,77,170)(40,149,78,171)(41,150,79,172)(42,151,80,173)(43,152,81,174)(44,153,82,175)(45,154,83,176)(46,155,84,177), (1,80,69,42)(2,81,47,43)(3,82,48,44)(4,83,49,45)(5,84,50,46)(6,85,51,24)(7,86,52,25)(8,87,53,26)(9,88,54,27)(10,89,55,28)(11,90,56,29)(12,91,57,30)(13,92,58,31)(14,70,59,32)(15,71,60,33)(16,72,61,34)(17,73,62,35)(18,74,63,36)(19,75,64,37)(20,76,65,38)(21,77,66,39)(22,78,67,40)(23,79,68,41)(93,148,131,170)(94,149,132,171)(95,150,133,172)(96,151,134,173)(97,152,135,174)(98,153,136,175)(99,154,137,176)(100,155,138,177)(101,156,116,178)(102,157,117,179)(103,158,118,180)(104,159,119,181)(105,160,120,182)(106,161,121,183)(107,139,122,184)(108,140,123,162)(109,141,124,163)(110,142,125,164)(111,143,126,165)(112,144,127,166)(113,145,128,167)(114,146,129,168)(115,147,130,169) );
G=PermutationGroup([[(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)], [(1,96,69,134),(2,97,47,135),(3,98,48,136),(4,99,49,137),(5,100,50,138),(6,101,51,116),(7,102,52,117),(8,103,53,118),(9,104,54,119),(10,105,55,120),(11,106,56,121),(12,107,57,122),(13,108,58,123),(14,109,59,124),(15,110,60,125),(16,111,61,126),(17,112,62,127),(18,113,63,128),(19,114,64,129),(20,115,65,130),(21,93,66,131),(22,94,67,132),(23,95,68,133),(24,156,85,178),(25,157,86,179),(26,158,87,180),(27,159,88,181),(28,160,89,182),(29,161,90,183),(30,139,91,184),(31,140,92,162),(32,141,70,163),(33,142,71,164),(34,143,72,165),(35,144,73,166),(36,145,74,167),(37,146,75,168),(38,147,76,169),(39,148,77,170),(40,149,78,171),(41,150,79,172),(42,151,80,173),(43,152,81,174),(44,153,82,175),(45,154,83,176),(46,155,84,177)], [(1,80,69,42),(2,81,47,43),(3,82,48,44),(4,83,49,45),(5,84,50,46),(6,85,51,24),(7,86,52,25),(8,87,53,26),(9,88,54,27),(10,89,55,28),(11,90,56,29),(12,91,57,30),(13,92,58,31),(14,70,59,32),(15,71,60,33),(16,72,61,34),(17,73,62,35),(18,74,63,36),(19,75,64,37),(20,76,65,38),(21,77,66,39),(22,78,67,40),(23,79,68,41),(93,148,131,170),(94,149,132,171),(95,150,133,172),(96,151,134,173),(97,152,135,174),(98,153,136,175),(99,154,137,176),(100,155,138,177),(101,156,116,178),(102,157,117,179),(103,158,118,180),(104,159,119,181),(105,160,120,182),(106,161,121,183),(107,139,122,184),(108,140,123,162),(109,141,124,163),(110,142,125,164),(111,143,126,165),(112,144,127,166),(113,145,128,167),(114,146,129,168),(115,147,130,169)]])
Q8×C23 is a maximal subgroup of
Q8⋊D23 C23⋊Q16 D92⋊C2
115 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 23A | ··· | 23V | 46A | ··· | 46V | 92A | ··· | 92BN |
| order | 1 | 2 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C23 | C46 | Q8 | Q8×C23 |
| kernel | Q8×C23 | C92 | Q8 | C4 | C23 | C1 |
| # reps | 1 | 3 | 22 | 66 | 1 | 22 |
Matrix representation of Q8×C23 ►in GL2(𝔽47) generated by
| 4 | 0 |
| 0 | 4 |
| 44 | 16 |
| 17 | 3 |
| 0 | 24 |
| 45 | 0 |
G:=sub<GL(2,GF(47))| [4,0,0,4],[44,17,16,3],[0,45,24,0] >;
Q8×C23 in GAP, Magma, Sage, TeX
Q_8\times C_{23} % in TeX
G:=Group("Q8xC23"); // GroupNames label
G:=SmallGroup(184,10);
// by ID
G=gap.SmallGroup(184,10);
# by ID
G:=PCGroup([4,-2,-2,-23,-2,368,753,373]);
// Polycyclic
G:=Group<a,b,c|a^23=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export