direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8⋊2C8, C8⋊2C24, C24⋊6C8, C12.56SD16, C12.31M4(2), C4⋊C8.4C6, (C4×C8).11C6, C4.6(C2×C24), C6.11(C4⋊C8), (C2×C8).10C12, C12.46(C2×C8), (C2×C24).13C4, (C4×C24).29C2, (C2×C12).47Q8, C6.8(C4.Q8), (C2×C12).530D4, C42.64(C2×C6), C4.15(C3×SD16), C4.4(C3×M4(2)), C6.7(C8.C4), (C4×C12).348C22, C2.3(C3×C4⋊C8), (C3×C4⋊C8).10C2, (C2×C4).8(C3×Q8), C2.1(C3×C4.Q8), (C2×C6).56(C4⋊C4), (C2×C4).61(C2×C12), C2.1(C3×C8.C4), (C2×C4).139(C3×D4), C22.13(C3×C4⋊C4), (C2×C12).322(C2×C4), SmallGroup(192,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊2C8
G = < a,b,c | a3=b8=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C3×C8⋊2C8
►Regular action on 192 points
Generators in S192
(1 66 40)(2 67 33)(3 68 34)(4 69 35)(5 70 36)(6 71 37)(7 72 38)(8 65 39)(9 192 184)(10 185 177)(11 186 178)(12 187 179)(13 188 180)(14 189 181)(15 190 182)(16 191 183)(17 77 30)(18 78 31)(19 79 32)(20 80 25)(21 73 26)(22 74 27)(23 75 28)(24 76 29)(41 97 111)(42 98 112)(43 99 105)(44 100 106)(45 101 107)(46 102 108)(47 103 109)(48 104 110)(49 134 59)(50 135 60)(51 136 61)(52 129 62)(53 130 63)(54 131 64)(55 132 57)(56 133 58)(81 147 91)(82 148 92)(83 149 93)(84 150 94)(85 151 95)(86 152 96)(87 145 89)(88 146 90)(113 141 155)(114 142 156)(115 143 157)(116 144 158)(117 137 159)(118 138 160)(119 139 153)(120 140 154)(121 174 166)(122 175 167)(123 176 168)(124 169 161)(125 170 162)(126 171 163)(127 172 164)(128 173 165)
(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)
(1 179 87 140 28 132 46 165)(2 182 88 143 29 135 47 168)(3 177 81 138 30 130 48 163)(4 180 82 141 31 133 41 166)(5 183 83 144 32 136 42 161)(6 178 84 139 25 131 43 164)(7 181 85 142 26 134 44 167)(8 184 86 137 27 129 45 162)(9 152 159 22 62 101 125 65)(10 147 160 17 63 104 126 68)(11 150 153 20 64 99 127 71)(12 145 154 23 57 102 128 66)(13 148 155 18 58 97 121 69)(14 151 156 21 59 100 122 72)(15 146 157 24 60 103 123 67)(16 149 158 19 61 98 124 70)(33 190 90 115 76 50 109 176)(34 185 91 118 77 53 110 171)(35 188 92 113 78 56 111 174)(36 191 93 116 79 51 112 169)(37 186 94 119 80 54 105 172)(38 189 95 114 73 49 106 175)(39 192 96 117 74 52 107 170)(40 187 89 120 75 55 108 173)
G:=sub<Sym(192)| (1,66,40)(2,67,33)(3,68,34)(4,69,35)(5,70,36)(6,71,37)(7,72,38)(8,65,39)(9,192,184)(10,185,177)(11,186,178)(12,187,179)(13,188,180)(14,189,181)(15,190,182)(16,191,183)(17,77,30)(18,78,31)(19,79,32)(20,80,25)(21,73,26)(22,74,27)(23,75,28)(24,76,29)(41,97,111)(42,98,112)(43,99,105)(44,100,106)(45,101,107)(46,102,108)(47,103,109)(48,104,110)(49,134,59)(50,135,60)(51,136,61)(52,129,62)(53,130,63)(54,131,64)(55,132,57)(56,133,58)(81,147,91)(82,148,92)(83,149,93)(84,150,94)(85,151,95)(86,152,96)(87,145,89)(88,146,90)(113,141,155)(114,142,156)(115,143,157)(116,144,158)(117,137,159)(118,138,160)(119,139,153)(120,140,154)(121,174,166)(122,175,167)(123,176,168)(124,169,161)(125,170,162)(126,171,163)(127,172,164)(128,173,165), (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), (1,179,87,140,28,132,46,165)(2,182,88,143,29,135,47,168)(3,177,81,138,30,130,48,163)(4,180,82,141,31,133,41,166)(5,183,83,144,32,136,42,161)(6,178,84,139,25,131,43,164)(7,181,85,142,26,134,44,167)(8,184,86,137,27,129,45,162)(9,152,159,22,62,101,125,65)(10,147,160,17,63,104,126,68)(11,150,153,20,64,99,127,71)(12,145,154,23,57,102,128,66)(13,148,155,18,58,97,121,69)(14,151,156,21,59,100,122,72)(15,146,157,24,60,103,123,67)(16,149,158,19,61,98,124,70)(33,190,90,115,76,50,109,176)(34,185,91,118,77,53,110,171)(35,188,92,113,78,56,111,174)(36,191,93,116,79,51,112,169)(37,186,94,119,80,54,105,172)(38,189,95,114,73,49,106,175)(39,192,96,117,74,52,107,170)(40,187,89,120,75,55,108,173)>;
G:=Group( (1,66,40)(2,67,33)(3,68,34)(4,69,35)(5,70,36)(6,71,37)(7,72,38)(8,65,39)(9,192,184)(10,185,177)(11,186,178)(12,187,179)(13,188,180)(14,189,181)(15,190,182)(16,191,183)(17,77,30)(18,78,31)(19,79,32)(20,80,25)(21,73,26)(22,74,27)(23,75,28)(24,76,29)(41,97,111)(42,98,112)(43,99,105)(44,100,106)(45,101,107)(46,102,108)(47,103,109)(48,104,110)(49,134,59)(50,135,60)(51,136,61)(52,129,62)(53,130,63)(54,131,64)(55,132,57)(56,133,58)(81,147,91)(82,148,92)(83,149,93)(84,150,94)(85,151,95)(86,152,96)(87,145,89)(88,146,90)(113,141,155)(114,142,156)(115,143,157)(116,144,158)(117,137,159)(118,138,160)(119,139,153)(120,140,154)(121,174,166)(122,175,167)(123,176,168)(124,169,161)(125,170,162)(126,171,163)(127,172,164)(128,173,165), (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), (1,179,87,140,28,132,46,165)(2,182,88,143,29,135,47,168)(3,177,81,138,30,130,48,163)(4,180,82,141,31,133,41,166)(5,183,83,144,32,136,42,161)(6,178,84,139,25,131,43,164)(7,181,85,142,26,134,44,167)(8,184,86,137,27,129,45,162)(9,152,159,22,62,101,125,65)(10,147,160,17,63,104,126,68)(11,150,153,20,64,99,127,71)(12,145,154,23,57,102,128,66)(13,148,155,18,58,97,121,69)(14,151,156,21,59,100,122,72)(15,146,157,24,60,103,123,67)(16,149,158,19,61,98,124,70)(33,190,90,115,76,50,109,176)(34,185,91,118,77,53,110,171)(35,188,92,113,78,56,111,174)(36,191,93,116,79,51,112,169)(37,186,94,119,80,54,105,172)(38,189,95,114,73,49,106,175)(39,192,96,117,74,52,107,170)(40,187,89,120,75,55,108,173) );
G=PermutationGroup([[(1,66,40),(2,67,33),(3,68,34),(4,69,35),(5,70,36),(6,71,37),(7,72,38),(8,65,39),(9,192,184),(10,185,177),(11,186,178),(12,187,179),(13,188,180),(14,189,181),(15,190,182),(16,191,183),(17,77,30),(18,78,31),(19,79,32),(20,80,25),(21,73,26),(22,74,27),(23,75,28),(24,76,29),(41,97,111),(42,98,112),(43,99,105),(44,100,106),(45,101,107),(46,102,108),(47,103,109),(48,104,110),(49,134,59),(50,135,60),(51,136,61),(52,129,62),(53,130,63),(54,131,64),(55,132,57),(56,133,58),(81,147,91),(82,148,92),(83,149,93),(84,150,94),(85,151,95),(86,152,96),(87,145,89),(88,146,90),(113,141,155),(114,142,156),(115,143,157),(116,144,158),(117,137,159),(118,138,160),(119,139,153),(120,140,154),(121,174,166),(122,175,167),(123,176,168),(124,169,161),(125,170,162),(126,171,163),(127,172,164),(128,173,165)], [(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)], [(1,179,87,140,28,132,46,165),(2,182,88,143,29,135,47,168),(3,177,81,138,30,130,48,163),(4,180,82,141,31,133,41,166),(5,183,83,144,32,136,42,161),(6,178,84,139,25,131,43,164),(7,181,85,142,26,134,44,167),(8,184,86,137,27,129,45,162),(9,152,159,22,62,101,125,65),(10,147,160,17,63,104,126,68),(11,150,153,20,64,99,127,71),(12,145,154,23,57,102,128,66),(13,148,155,18,58,97,121,69),(14,151,156,21,59,100,122,72),(15,146,157,24,60,103,123,67),(16,149,158,19,61,98,124,70),(33,190,90,115,76,50,109,176),(34,185,91,118,77,53,110,171),(35,188,92,113,78,56,111,174),(36,191,93,116,79,51,112,169),(37,186,94,119,80,54,105,172),(38,189,95,114,73,49,106,175),(39,192,96,117,74,52,107,170),(40,187,89,120,75,55,108,173)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 12I | ··· | 12P | 24A | ··· | 24P | 24Q | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C8 | C12 | C24 | D4 | Q8 | M4(2) | SD16 | C3×D4 | C3×Q8 | C8.C4 | C3×M4(2) | C3×SD16 | C3×C8.C4 |
| kernel | C3×C8⋊2C8 | C4×C24 | C3×C4⋊C8 | C8⋊2C8 | C2×C24 | C4×C8 | C4⋊C8 | C24 | C2×C8 | C8 | C2×C12 | C2×C12 | C12 | C12 | C2×C4 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 8 | 16 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C8⋊2C8 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 2 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 67 | 6 |
| 0 | 0 | 67 | 67 |
| 20 | 18 | 0 | 0 |
| 29 | 53 | 0 | 0 |
| 0 | 0 | 43 | 52 |
| 0 | 0 | 52 | 30 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[72,72,0,0,2,1,0,0,0,0,67,67,0,0,6,67],[20,29,0,0,18,53,0,0,0,0,43,52,0,0,52,30] >;
C3×C8⋊2C8 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes_2C_8
% in TeX
G:=Group("C3xC8:2C8"); // GroupNames label
G:=SmallGroup(192,140);
// by ID
G=gap.SmallGroup(192,140);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,92,1683,136,172]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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