direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C6×Q32, C24.70D4, C12.46D8, C48.22C22, C24.65C23, C4.8(C3×D8), C4.9(C6×D4), (C2×C16).4C6, C16.5(C2×C6), C2.14(C6×D8), (C2×C6).57D8, C8.11(C3×D4), C6.86(C2×D8), (C2×C48).10C2, C8.5(C22×C6), Q16.1(C2×C6), (C2×Q16).4C6, (C2×C12).428D4, C12.316(C2×D4), (C6×Q16).11C2, C22.16(C3×D8), (C2×C24).406C22, (C3×Q16).13C22, (C2×C8).86(C2×C6), (C2×C4).84(C3×D4), SmallGroup(192,940)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Q32
G = < a,b,c | a6=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 146 in 82 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C16, C2×C8, Q16, Q16, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C2×C16, Q32, C2×Q16, C48, C2×C24, C3×Q16, C3×Q16, C6×Q8, C2×Q32, C2×C48, C3×Q32, C6×Q16, C6×Q32
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, Q32, C2×D8, C3×D8, C6×D4, C2×Q32, C3×Q32, C6×D8, C6×Q32
►Regular action on 192 points
Generators in S192
(1 151 32 41 111 176)(2 152 17 42 112 161)(3 153 18 43 97 162)(4 154 19 44 98 163)(5 155 20 45 99 164)(6 156 21 46 100 165)(7 157 22 47 101 166)(8 158 23 48 102 167)(9 159 24 33 103 168)(10 160 25 34 104 169)(11 145 26 35 105 170)(12 146 27 36 106 171)(13 147 28 37 107 172)(14 148 29 38 108 173)(15 149 30 39 109 174)(16 150 31 40 110 175)(49 127 181 78 81 139)(50 128 182 79 82 140)(51 113 183 80 83 141)(52 114 184 65 84 142)(53 115 185 66 85 143)(54 116 186 67 86 144)(55 117 187 68 87 129)(56 118 188 69 88 130)(57 119 189 70 89 131)(58 120 190 71 90 132)(59 121 191 72 91 133)(60 122 192 73 92 134)(61 123 177 74 93 135)(62 124 178 75 94 136)(63 125 179 76 95 137)(64 126 180 77 96 138)
(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)(177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)
(1 68 9 76)(2 67 10 75)(3 66 11 74)(4 65 12 73)(5 80 13 72)(6 79 14 71)(7 78 15 70)(8 77 16 69)(17 144 25 136)(18 143 26 135)(19 142 27 134)(20 141 28 133)(21 140 29 132)(22 139 30 131)(23 138 31 130)(24 137 32 129)(33 63 41 55)(34 62 42 54)(35 61 43 53)(36 60 44 52)(37 59 45 51)(38 58 46 50)(39 57 47 49)(40 56 48 64)(81 149 89 157)(82 148 90 156)(83 147 91 155)(84 146 92 154)(85 145 93 153)(86 160 94 152)(87 159 95 151)(88 158 96 150)(97 115 105 123)(98 114 106 122)(99 113 107 121)(100 128 108 120)(101 127 109 119)(102 126 110 118)(103 125 111 117)(104 124 112 116)(161 186 169 178)(162 185 170 177)(163 184 171 192)(164 183 172 191)(165 182 173 190)(166 181 174 189)(167 180 175 188)(168 179 176 187)
G:=sub<Sym(192)| (1,151,32,41,111,176)(2,152,17,42,112,161)(3,153,18,43,97,162)(4,154,19,44,98,163)(5,155,20,45,99,164)(6,156,21,46,100,165)(7,157,22,47,101,166)(8,158,23,48,102,167)(9,159,24,33,103,168)(10,160,25,34,104,169)(11,145,26,35,105,170)(12,146,27,36,106,171)(13,147,28,37,107,172)(14,148,29,38,108,173)(15,149,30,39,109,174)(16,150,31,40,110,175)(49,127,181,78,81,139)(50,128,182,79,82,140)(51,113,183,80,83,141)(52,114,184,65,84,142)(53,115,185,66,85,143)(54,116,186,67,86,144)(55,117,187,68,87,129)(56,118,188,69,88,130)(57,119,189,70,89,131)(58,120,190,71,90,132)(59,121,191,72,91,133)(60,122,192,73,92,134)(61,123,177,74,93,135)(62,124,178,75,94,136)(63,125,179,76,95,137)(64,126,180,77,96,138), (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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,68,9,76)(2,67,10,75)(3,66,11,74)(4,65,12,73)(5,80,13,72)(6,79,14,71)(7,78,15,70)(8,77,16,69)(17,144,25,136)(18,143,26,135)(19,142,27,134)(20,141,28,133)(21,140,29,132)(22,139,30,131)(23,138,31,130)(24,137,32,129)(33,63,41,55)(34,62,42,54)(35,61,43,53)(36,60,44,52)(37,59,45,51)(38,58,46,50)(39,57,47,49)(40,56,48,64)(81,149,89,157)(82,148,90,156)(83,147,91,155)(84,146,92,154)(85,145,93,153)(86,160,94,152)(87,159,95,151)(88,158,96,150)(97,115,105,123)(98,114,106,122)(99,113,107,121)(100,128,108,120)(101,127,109,119)(102,126,110,118)(103,125,111,117)(104,124,112,116)(161,186,169,178)(162,185,170,177)(163,184,171,192)(164,183,172,191)(165,182,173,190)(166,181,174,189)(167,180,175,188)(168,179,176,187)>;
G:=Group( (1,151,32,41,111,176)(2,152,17,42,112,161)(3,153,18,43,97,162)(4,154,19,44,98,163)(5,155,20,45,99,164)(6,156,21,46,100,165)(7,157,22,47,101,166)(8,158,23,48,102,167)(9,159,24,33,103,168)(10,160,25,34,104,169)(11,145,26,35,105,170)(12,146,27,36,106,171)(13,147,28,37,107,172)(14,148,29,38,108,173)(15,149,30,39,109,174)(16,150,31,40,110,175)(49,127,181,78,81,139)(50,128,182,79,82,140)(51,113,183,80,83,141)(52,114,184,65,84,142)(53,115,185,66,85,143)(54,116,186,67,86,144)(55,117,187,68,87,129)(56,118,188,69,88,130)(57,119,189,70,89,131)(58,120,190,71,90,132)(59,121,191,72,91,133)(60,122,192,73,92,134)(61,123,177,74,93,135)(62,124,178,75,94,136)(63,125,179,76,95,137)(64,126,180,77,96,138), (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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,68,9,76)(2,67,10,75)(3,66,11,74)(4,65,12,73)(5,80,13,72)(6,79,14,71)(7,78,15,70)(8,77,16,69)(17,144,25,136)(18,143,26,135)(19,142,27,134)(20,141,28,133)(21,140,29,132)(22,139,30,131)(23,138,31,130)(24,137,32,129)(33,63,41,55)(34,62,42,54)(35,61,43,53)(36,60,44,52)(37,59,45,51)(38,58,46,50)(39,57,47,49)(40,56,48,64)(81,149,89,157)(82,148,90,156)(83,147,91,155)(84,146,92,154)(85,145,93,153)(86,160,94,152)(87,159,95,151)(88,158,96,150)(97,115,105,123)(98,114,106,122)(99,113,107,121)(100,128,108,120)(101,127,109,119)(102,126,110,118)(103,125,111,117)(104,124,112,116)(161,186,169,178)(162,185,170,177)(163,184,171,192)(164,183,172,191)(165,182,173,190)(166,181,174,189)(167,180,175,188)(168,179,176,187) );
G=PermutationGroup([[(1,151,32,41,111,176),(2,152,17,42,112,161),(3,153,18,43,97,162),(4,154,19,44,98,163),(5,155,20,45,99,164),(6,156,21,46,100,165),(7,157,22,47,101,166),(8,158,23,48,102,167),(9,159,24,33,103,168),(10,160,25,34,104,169),(11,145,26,35,105,170),(12,146,27,36,106,171),(13,147,28,37,107,172),(14,148,29,38,108,173),(15,149,30,39,109,174),(16,150,31,40,110,175),(49,127,181,78,81,139),(50,128,182,79,82,140),(51,113,183,80,83,141),(52,114,184,65,84,142),(53,115,185,66,85,143),(54,116,186,67,86,144),(55,117,187,68,87,129),(56,118,188,69,88,130),(57,119,189,70,89,131),(58,120,190,71,90,132),(59,121,191,72,91,133),(60,122,192,73,92,134),(61,123,177,74,93,135),(62,124,178,75,94,136),(63,125,179,76,95,137),(64,126,180,77,96,138)], [(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),(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)], [(1,68,9,76),(2,67,10,75),(3,66,11,74),(4,65,12,73),(5,80,13,72),(6,79,14,71),(7,78,15,70),(8,77,16,69),(17,144,25,136),(18,143,26,135),(19,142,27,134),(20,141,28,133),(21,140,29,132),(22,139,30,131),(23,138,31,130),(24,137,32,129),(33,63,41,55),(34,62,42,54),(35,61,43,53),(36,60,44,52),(37,59,45,51),(38,58,46,50),(39,57,47,49),(40,56,48,64),(81,149,89,157),(82,148,90,156),(83,147,91,155),(84,146,92,154),(85,145,93,153),(86,160,94,152),(87,159,95,151),(88,158,96,150),(97,115,105,123),(98,114,106,122),(99,113,107,121),(100,128,108,120),(101,127,109,119),(102,126,110,118),(103,125,111,117),(104,124,112,116),(161,186,169,178),(162,185,170,177),(163,184,171,192),(164,183,172,191),(165,182,173,190),(166,181,174,189),(167,180,175,188),(168,179,176,187)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | D8 | D8 | C3×D4 | C3×D4 | Q32 | C3×D8 | C3×D8 | C3×Q32 |
| kernel | C6×Q32 | C2×C48 | C3×Q32 | C6×Q16 | C2×Q32 | C2×C16 | Q32 | C2×Q16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C6×Q32 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 1 | 0 | 0 |
| 0 | 71 | 2 |
| 0 | 95 | 71 |
| 96 | 0 | 0 |
| 0 | 31 | 28 |
| 0 | 28 | 66 |
G:=sub<GL(3,GF(97))| [96,0,0,0,61,0,0,0,61],[1,0,0,0,71,95,0,2,71],[96,0,0,0,31,28,0,28,66] >;
C6×Q32 in GAP, Magma, Sage, TeX
C_6\times Q_{32} % in TeX
G:=Group("C6xQ32"); // GroupNames label
G:=SmallGroup(192,940);
// by ID
G=gap.SmallGroup(192,940);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,680,2524,1271,242,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c|a^6=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations