direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C8⋊C8, C24⋊7C8, C8⋊3C24, C12.36M4(2), C2.1(C4×C24), (C2×C8).4C12, C6.10(C4×C8), (C4×C8).13C6, (C4×C24).31C2, C12.51(C2×C8), C4.11(C2×C24), (C2×C24).30C4, C6.7(C8⋊C4), C42.91(C2×C6), C22.6(C4×C12), (C2×C6).28C42, C4.9(C3×M4(2)), (C4×C12).377C22, C2.1(C3×C8⋊C4), (C2×C4).80(C2×C12), (C2×C12).342(C2×C4), SmallGroup(192,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊C8
G = < a,b,c | a3=b8=c8=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 74 in 66 conjugacy classes, 58 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, C12, C2×C6, C42, C2×C8, C24, C24, C2×C12, C2×C12, C4×C8, C4×C8, C4×C12, C2×C24, C8⋊C8, C4×C24, C4×C24, C3×C8⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C42, C2×C8, M4(2), C24, C2×C12, C4×C8, C8⋊C4, C4×C12, C2×C24, C3×M4(2), C8⋊C8, C4×C24, C3×C8⋊C4, C3×C8⋊C8
►Regular action on 192 points
Generators in S192
(1 173 71)(2 174 72)(3 175 65)(4 176 66)(5 169 67)(6 170 68)(7 171 69)(8 172 70)(9 96 108)(10 89 109)(11 90 110)(12 91 111)(13 92 112)(14 93 105)(15 94 106)(16 95 107)(17 33 127)(18 34 128)(19 35 121)(20 36 122)(21 37 123)(22 38 124)(23 39 125)(24 40 126)(25 117 133)(26 118 134)(27 119 135)(28 120 136)(29 113 129)(30 114 130)(31 115 131)(32 116 132)(41 57 149)(42 58 150)(43 59 151)(44 60 152)(45 61 145)(46 62 146)(47 63 147)(48 64 148)(49 143 159)(50 144 160)(51 137 153)(52 138 154)(53 139 155)(54 140 156)(55 141 157)(56 142 158)(73 165 181)(74 166 182)(75 167 183)(76 168 184)(77 161 177)(78 162 178)(79 163 179)(80 164 180)(81 101 191)(82 102 192)(83 103 185)(84 104 186)(85 97 187)(86 98 188)(87 99 189)(88 100 190)
(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)
(1 143 17 90 166 41 117 87)(2 140 18 95 167 46 118 84)(3 137 19 92 168 43 119 81)(4 142 20 89 161 48 120 86)(5 139 21 94 162 45 113 83)(6 144 22 91 163 42 114 88)(7 141 23 96 164 47 115 85)(8 138 24 93 165 44 116 82)(9 80 147 31 187 69 55 125)(10 77 148 28 188 66 56 122)(11 74 149 25 189 71 49 127)(12 79 150 30 190 68 50 124)(13 76 151 27 191 65 51 121)(14 73 152 32 192 70 52 126)(15 78 145 29 185 67 53 123)(16 75 146 26 186 72 54 128)(33 110 182 57 133 99 173 159)(34 107 183 62 134 104 174 156)(35 112 184 59 135 101 175 153)(36 109 177 64 136 98 176 158)(37 106 178 61 129 103 169 155)(38 111 179 58 130 100 170 160)(39 108 180 63 131 97 171 157)(40 105 181 60 132 102 172 154)
G:=sub<Sym(192)| (1,173,71)(2,174,72)(3,175,65)(4,176,66)(5,169,67)(6,170,68)(7,171,69)(8,172,70)(9,96,108)(10,89,109)(11,90,110)(12,91,111)(13,92,112)(14,93,105)(15,94,106)(16,95,107)(17,33,127)(18,34,128)(19,35,121)(20,36,122)(21,37,123)(22,38,124)(23,39,125)(24,40,126)(25,117,133)(26,118,134)(27,119,135)(28,120,136)(29,113,129)(30,114,130)(31,115,131)(32,116,132)(41,57,149)(42,58,150)(43,59,151)(44,60,152)(45,61,145)(46,62,146)(47,63,147)(48,64,148)(49,143,159)(50,144,160)(51,137,153)(52,138,154)(53,139,155)(54,140,156)(55,141,157)(56,142,158)(73,165,181)(74,166,182)(75,167,183)(76,168,184)(77,161,177)(78,162,178)(79,163,179)(80,164,180)(81,101,191)(82,102,192)(83,103,185)(84,104,186)(85,97,187)(86,98,188)(87,99,189)(88,100,190), (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), (1,143,17,90,166,41,117,87)(2,140,18,95,167,46,118,84)(3,137,19,92,168,43,119,81)(4,142,20,89,161,48,120,86)(5,139,21,94,162,45,113,83)(6,144,22,91,163,42,114,88)(7,141,23,96,164,47,115,85)(8,138,24,93,165,44,116,82)(9,80,147,31,187,69,55,125)(10,77,148,28,188,66,56,122)(11,74,149,25,189,71,49,127)(12,79,150,30,190,68,50,124)(13,76,151,27,191,65,51,121)(14,73,152,32,192,70,52,126)(15,78,145,29,185,67,53,123)(16,75,146,26,186,72,54,128)(33,110,182,57,133,99,173,159)(34,107,183,62,134,104,174,156)(35,112,184,59,135,101,175,153)(36,109,177,64,136,98,176,158)(37,106,178,61,129,103,169,155)(38,111,179,58,130,100,170,160)(39,108,180,63,131,97,171,157)(40,105,181,60,132,102,172,154)>;
G:=Group( (1,173,71)(2,174,72)(3,175,65)(4,176,66)(5,169,67)(6,170,68)(7,171,69)(8,172,70)(9,96,108)(10,89,109)(11,90,110)(12,91,111)(13,92,112)(14,93,105)(15,94,106)(16,95,107)(17,33,127)(18,34,128)(19,35,121)(20,36,122)(21,37,123)(22,38,124)(23,39,125)(24,40,126)(25,117,133)(26,118,134)(27,119,135)(28,120,136)(29,113,129)(30,114,130)(31,115,131)(32,116,132)(41,57,149)(42,58,150)(43,59,151)(44,60,152)(45,61,145)(46,62,146)(47,63,147)(48,64,148)(49,143,159)(50,144,160)(51,137,153)(52,138,154)(53,139,155)(54,140,156)(55,141,157)(56,142,158)(73,165,181)(74,166,182)(75,167,183)(76,168,184)(77,161,177)(78,162,178)(79,163,179)(80,164,180)(81,101,191)(82,102,192)(83,103,185)(84,104,186)(85,97,187)(86,98,188)(87,99,189)(88,100,190), (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), (1,143,17,90,166,41,117,87)(2,140,18,95,167,46,118,84)(3,137,19,92,168,43,119,81)(4,142,20,89,161,48,120,86)(5,139,21,94,162,45,113,83)(6,144,22,91,163,42,114,88)(7,141,23,96,164,47,115,85)(8,138,24,93,165,44,116,82)(9,80,147,31,187,69,55,125)(10,77,148,28,188,66,56,122)(11,74,149,25,189,71,49,127)(12,79,150,30,190,68,50,124)(13,76,151,27,191,65,51,121)(14,73,152,32,192,70,52,126)(15,78,145,29,185,67,53,123)(16,75,146,26,186,72,54,128)(33,110,182,57,133,99,173,159)(34,107,183,62,134,104,174,156)(35,112,184,59,135,101,175,153)(36,109,177,64,136,98,176,158)(37,106,178,61,129,103,169,155)(38,111,179,58,130,100,170,160)(39,108,180,63,131,97,171,157)(40,105,181,60,132,102,172,154) );
G=PermutationGroup([[(1,173,71),(2,174,72),(3,175,65),(4,176,66),(5,169,67),(6,170,68),(7,171,69),(8,172,70),(9,96,108),(10,89,109),(11,90,110),(12,91,111),(13,92,112),(14,93,105),(15,94,106),(16,95,107),(17,33,127),(18,34,128),(19,35,121),(20,36,122),(21,37,123),(22,38,124),(23,39,125),(24,40,126),(25,117,133),(26,118,134),(27,119,135),(28,120,136),(29,113,129),(30,114,130),(31,115,131),(32,116,132),(41,57,149),(42,58,150),(43,59,151),(44,60,152),(45,61,145),(46,62,146),(47,63,147),(48,64,148),(49,143,159),(50,144,160),(51,137,153),(52,138,154),(53,139,155),(54,140,156),(55,141,157),(56,142,158),(73,165,181),(74,166,182),(75,167,183),(76,168,184),(77,161,177),(78,162,178),(79,163,179),(80,164,180),(81,101,191),(82,102,192),(83,103,185),(84,104,186),(85,97,187),(86,98,188),(87,99,189),(88,100,190)], [(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)], [(1,143,17,90,166,41,117,87),(2,140,18,95,167,46,118,84),(3,137,19,92,168,43,119,81),(4,142,20,89,161,48,120,86),(5,139,21,94,162,45,113,83),(6,144,22,91,163,42,114,88),(7,141,23,96,164,47,115,85),(8,138,24,93,165,44,116,82),(9,80,147,31,187,69,55,125),(10,77,148,28,188,66,56,122),(11,74,149,25,189,71,49,127),(12,79,150,30,190,68,50,124),(13,76,151,27,191,65,51,121),(14,73,152,32,192,70,52,126),(15,78,145,29,185,67,53,123),(16,75,146,26,186,72,54,128),(33,110,182,57,133,99,173,159),(34,107,183,62,134,104,174,156),(35,112,184,59,135,101,175,153),(36,109,177,64,136,98,176,158),(37,106,178,61,129,103,169,155),(38,111,179,58,130,100,170,160),(39,108,180,63,131,97,171,157),(40,105,181,60,132,102,172,154)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 8A | ··· | 8X | 12A | ··· | 12X | 24A | ··· | 24AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | M4(2) | C3×M4(2) |
| kernel | C3×C8⋊C8 | C4×C24 | C8⋊C8 | C2×C24 | C4×C8 | C24 | C2×C8 | C8 | C12 | C4 |
| # reps | 1 | 3 | 2 | 12 | 6 | 16 | 24 | 32 | 8 | 16 |
Matrix representation of C3×C8⋊C8 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 46 | 0 | 0 |
| 0 | 30 | 9 |
| 0 | 43 | 43 |
| 51 | 0 | 0 |
| 0 | 72 | 71 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[46,0,0,0,30,43,0,9,43],[51,0,0,0,72,0,0,71,1] >;
C3×C8⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes C_8
% in TeX
G:=Group("C3xC8:C8"); // GroupNames label
G:=SmallGroup(192,128);
// by ID
G=gap.SmallGroup(192,128);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,701,176,136,172]);
// 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^5>;
// generators/relations