direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×Q8⋊C8, Q8⋊2C24, C12.30Q16, C12.55SD16, C12.20M4(2), C4⋊C8.1C6, (C4×C8).1C6, (C3×Q8)⋊3C8, C6.21C4≀C2, C4⋊C4.4C12, C4.2(C2×C24), (C4×C24).3C2, (C4×Q8).7C6, C4.8(C3×Q16), C12.31(C2×C8), (C6×Q8).14C4, (C2×Q8).7C12, (C2×C12).529D4, C42.63(C2×C6), (Q8×C12).14C2, C4.14(C3×SD16), C4.2(C3×M4(2)), C6.23(C22⋊C8), (C4×C12).347C22, C6.14(Q8⋊C4), C2.2(C3×C4≀C2), (C3×C4⋊C8).7C2, (C3×C4⋊C4).16C4, (C2×C4).94(C3×D4), C2.6(C3×C22⋊C8), (C2×C4).39(C2×C12), C2.1(C3×Q8⋊C4), (C2×C12).259(C2×C4), C22.26(C3×C22⋊C4), (C2×C6).121(C22⋊C4), SmallGroup(192,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8⋊C8
G = < a,b,c,d | a3=b4=d8=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 106 in 70 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C4×C8, C4⋊C8, C4×Q8, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, Q8⋊C8, C4×C24, C3×C4⋊C8, Q8×C12, C3×Q8⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C8, M4(2), SD16, Q16, C24, C2×C12, C3×D4, C22⋊C8, Q8⋊C4, C4≀C2, C3×C22⋊C4, C2×C24, C3×M4(2), C3×SD16, C3×Q16, Q8⋊C8, C3×C22⋊C8, C3×Q8⋊C4, C3×C4≀C2, C3×Q8⋊C8
►Regular action on 192 points
Generators in S192
(1 121 113)(2 122 114)(3 123 115)(4 124 116)(5 125 117)(6 126 118)(7 127 119)(8 128 120)(9 138 130)(10 139 131)(11 140 132)(12 141 133)(13 142 134)(14 143 135)(15 144 136)(16 137 129)(17 33 25)(18 34 26)(19 35 27)(20 36 28)(21 37 29)(22 38 30)(23 39 31)(24 40 32)(41 109 49)(42 110 50)(43 111 51)(44 112 52)(45 105 53)(46 106 54)(47 107 55)(48 108 56)(57 73 65)(58 74 66)(59 75 67)(60 76 68)(61 77 69)(62 78 70)(63 79 71)(64 80 72)(81 103 89)(82 104 90)(83 97 91)(84 98 92)(85 99 93)(86 100 94)(87 101 95)(88 102 96)(145 161 153)(146 162 154)(147 163 155)(148 164 156)(149 165 157)(150 166 158)(151 167 159)(152 168 160)(169 185 177)(170 186 178)(171 187 179)(172 188 180)(173 189 181)(174 190 182)(175 191 183)(176 192 184)
(1 63 151 17)(2 18 152 64)(3 57 145 19)(4 20 146 58)(5 59 147 21)(6 22 148 60)(7 61 149 23)(8 24 150 62)(9 102 192 110)(10 111 185 103)(11 104 186 112)(12 105 187 97)(13 98 188 106)(14 107 189 99)(15 100 190 108)(16 109 191 101)(25 113 71 159)(26 160 72 114)(27 115 65 153)(28 154 66 116)(29 117 67 155)(30 156 68 118)(31 119 69 157)(32 158 70 120)(33 121 79 167)(34 168 80 122)(35 123 73 161)(36 162 74 124)(37 125 75 163)(38 164 76 126)(39 127 77 165)(40 166 78 128)(41 175 87 129)(42 130 88 176)(43 169 81 131)(44 132 82 170)(45 171 83 133)(46 134 84 172)(47 173 85 135)(48 136 86 174)(49 183 95 137)(50 138 96 184)(51 177 89 139)(52 140 90 178)(53 179 91 141)(54 142 92 180)(55 181 93 143)(56 144 94 182)
(1 171 151 133)(2 84 152 46)(3 173 145 135)(4 86 146 48)(5 175 147 129)(6 88 148 42)(7 169 149 131)(8 82 150 44)(9 76 192 38)(10 127 185 165)(11 78 186 40)(12 121 187 167)(13 80 188 34)(14 123 189 161)(15 74 190 36)(16 125 191 163)(17 83 63 45)(18 134 64 172)(19 85 57 47)(20 136 58 174)(21 87 59 41)(22 130 60 176)(23 81 61 43)(24 132 62 170)(25 91 71 53)(26 142 72 180)(27 93 65 55)(28 144 66 182)(29 95 67 49)(30 138 68 184)(31 89 69 51)(32 140 70 178)(33 97 79 105)(35 99 73 107)(37 101 75 109)(39 103 77 111)(50 118 96 156)(52 120 90 158)(54 114 92 160)(56 116 94 154)(98 168 106 122)(100 162 108 124)(102 164 110 126)(104 166 112 128)(113 179 159 141)(115 181 153 143)(117 183 155 137)(119 177 157 139)
(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)
G:=sub<Sym(192)| (1,121,113)(2,122,114)(3,123,115)(4,124,116)(5,125,117)(6,126,118)(7,127,119)(8,128,120)(9,138,130)(10,139,131)(11,140,132)(12,141,133)(13,142,134)(14,143,135)(15,144,136)(16,137,129)(17,33,25)(18,34,26)(19,35,27)(20,36,28)(21,37,29)(22,38,30)(23,39,31)(24,40,32)(41,109,49)(42,110,50)(43,111,51)(44,112,52)(45,105,53)(46,106,54)(47,107,55)(48,108,56)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(61,77,69)(62,78,70)(63,79,71)(64,80,72)(81,103,89)(82,104,90)(83,97,91)(84,98,92)(85,99,93)(86,100,94)(87,101,95)(88,102,96)(145,161,153)(146,162,154)(147,163,155)(148,164,156)(149,165,157)(150,166,158)(151,167,159)(152,168,160)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (1,63,151,17)(2,18,152,64)(3,57,145,19)(4,20,146,58)(5,59,147,21)(6,22,148,60)(7,61,149,23)(8,24,150,62)(9,102,192,110)(10,111,185,103)(11,104,186,112)(12,105,187,97)(13,98,188,106)(14,107,189,99)(15,100,190,108)(16,109,191,101)(25,113,71,159)(26,160,72,114)(27,115,65,153)(28,154,66,116)(29,117,67,155)(30,156,68,118)(31,119,69,157)(32,158,70,120)(33,121,79,167)(34,168,80,122)(35,123,73,161)(36,162,74,124)(37,125,75,163)(38,164,76,126)(39,127,77,165)(40,166,78,128)(41,175,87,129)(42,130,88,176)(43,169,81,131)(44,132,82,170)(45,171,83,133)(46,134,84,172)(47,173,85,135)(48,136,86,174)(49,183,95,137)(50,138,96,184)(51,177,89,139)(52,140,90,178)(53,179,91,141)(54,142,92,180)(55,181,93,143)(56,144,94,182), (1,171,151,133)(2,84,152,46)(3,173,145,135)(4,86,146,48)(5,175,147,129)(6,88,148,42)(7,169,149,131)(8,82,150,44)(9,76,192,38)(10,127,185,165)(11,78,186,40)(12,121,187,167)(13,80,188,34)(14,123,189,161)(15,74,190,36)(16,125,191,163)(17,83,63,45)(18,134,64,172)(19,85,57,47)(20,136,58,174)(21,87,59,41)(22,130,60,176)(23,81,61,43)(24,132,62,170)(25,91,71,53)(26,142,72,180)(27,93,65,55)(28,144,66,182)(29,95,67,49)(30,138,68,184)(31,89,69,51)(32,140,70,178)(33,97,79,105)(35,99,73,107)(37,101,75,109)(39,103,77,111)(50,118,96,156)(52,120,90,158)(54,114,92,160)(56,116,94,154)(98,168,106,122)(100,162,108,124)(102,164,110,126)(104,166,112,128)(113,179,159,141)(115,181,153,143)(117,183,155,137)(119,177,157,139), (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)>;
G:=Group( (1,121,113)(2,122,114)(3,123,115)(4,124,116)(5,125,117)(6,126,118)(7,127,119)(8,128,120)(9,138,130)(10,139,131)(11,140,132)(12,141,133)(13,142,134)(14,143,135)(15,144,136)(16,137,129)(17,33,25)(18,34,26)(19,35,27)(20,36,28)(21,37,29)(22,38,30)(23,39,31)(24,40,32)(41,109,49)(42,110,50)(43,111,51)(44,112,52)(45,105,53)(46,106,54)(47,107,55)(48,108,56)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(61,77,69)(62,78,70)(63,79,71)(64,80,72)(81,103,89)(82,104,90)(83,97,91)(84,98,92)(85,99,93)(86,100,94)(87,101,95)(88,102,96)(145,161,153)(146,162,154)(147,163,155)(148,164,156)(149,165,157)(150,166,158)(151,167,159)(152,168,160)(169,185,177)(170,186,178)(171,187,179)(172,188,180)(173,189,181)(174,190,182)(175,191,183)(176,192,184), (1,63,151,17)(2,18,152,64)(3,57,145,19)(4,20,146,58)(5,59,147,21)(6,22,148,60)(7,61,149,23)(8,24,150,62)(9,102,192,110)(10,111,185,103)(11,104,186,112)(12,105,187,97)(13,98,188,106)(14,107,189,99)(15,100,190,108)(16,109,191,101)(25,113,71,159)(26,160,72,114)(27,115,65,153)(28,154,66,116)(29,117,67,155)(30,156,68,118)(31,119,69,157)(32,158,70,120)(33,121,79,167)(34,168,80,122)(35,123,73,161)(36,162,74,124)(37,125,75,163)(38,164,76,126)(39,127,77,165)(40,166,78,128)(41,175,87,129)(42,130,88,176)(43,169,81,131)(44,132,82,170)(45,171,83,133)(46,134,84,172)(47,173,85,135)(48,136,86,174)(49,183,95,137)(50,138,96,184)(51,177,89,139)(52,140,90,178)(53,179,91,141)(54,142,92,180)(55,181,93,143)(56,144,94,182), (1,171,151,133)(2,84,152,46)(3,173,145,135)(4,86,146,48)(5,175,147,129)(6,88,148,42)(7,169,149,131)(8,82,150,44)(9,76,192,38)(10,127,185,165)(11,78,186,40)(12,121,187,167)(13,80,188,34)(14,123,189,161)(15,74,190,36)(16,125,191,163)(17,83,63,45)(18,134,64,172)(19,85,57,47)(20,136,58,174)(21,87,59,41)(22,130,60,176)(23,81,61,43)(24,132,62,170)(25,91,71,53)(26,142,72,180)(27,93,65,55)(28,144,66,182)(29,95,67,49)(30,138,68,184)(31,89,69,51)(32,140,70,178)(33,97,79,105)(35,99,73,107)(37,101,75,109)(39,103,77,111)(50,118,96,156)(52,120,90,158)(54,114,92,160)(56,116,94,154)(98,168,106,122)(100,162,108,124)(102,164,110,126)(104,166,112,128)(113,179,159,141)(115,181,153,143)(117,183,155,137)(119,177,157,139), (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) );
G=PermutationGroup([[(1,121,113),(2,122,114),(3,123,115),(4,124,116),(5,125,117),(6,126,118),(7,127,119),(8,128,120),(9,138,130),(10,139,131),(11,140,132),(12,141,133),(13,142,134),(14,143,135),(15,144,136),(16,137,129),(17,33,25),(18,34,26),(19,35,27),(20,36,28),(21,37,29),(22,38,30),(23,39,31),(24,40,32),(41,109,49),(42,110,50),(43,111,51),(44,112,52),(45,105,53),(46,106,54),(47,107,55),(48,108,56),(57,73,65),(58,74,66),(59,75,67),(60,76,68),(61,77,69),(62,78,70),(63,79,71),(64,80,72),(81,103,89),(82,104,90),(83,97,91),(84,98,92),(85,99,93),(86,100,94),(87,101,95),(88,102,96),(145,161,153),(146,162,154),(147,163,155),(148,164,156),(149,165,157),(150,166,158),(151,167,159),(152,168,160),(169,185,177),(170,186,178),(171,187,179),(172,188,180),(173,189,181),(174,190,182),(175,191,183),(176,192,184)], [(1,63,151,17),(2,18,152,64),(3,57,145,19),(4,20,146,58),(5,59,147,21),(6,22,148,60),(7,61,149,23),(8,24,150,62),(9,102,192,110),(10,111,185,103),(11,104,186,112),(12,105,187,97),(13,98,188,106),(14,107,189,99),(15,100,190,108),(16,109,191,101),(25,113,71,159),(26,160,72,114),(27,115,65,153),(28,154,66,116),(29,117,67,155),(30,156,68,118),(31,119,69,157),(32,158,70,120),(33,121,79,167),(34,168,80,122),(35,123,73,161),(36,162,74,124),(37,125,75,163),(38,164,76,126),(39,127,77,165),(40,166,78,128),(41,175,87,129),(42,130,88,176),(43,169,81,131),(44,132,82,170),(45,171,83,133),(46,134,84,172),(47,173,85,135),(48,136,86,174),(49,183,95,137),(50,138,96,184),(51,177,89,139),(52,140,90,178),(53,179,91,141),(54,142,92,180),(55,181,93,143),(56,144,94,182)], [(1,171,151,133),(2,84,152,46),(3,173,145,135),(4,86,146,48),(5,175,147,129),(6,88,148,42),(7,169,149,131),(8,82,150,44),(9,76,192,38),(10,127,185,165),(11,78,186,40),(12,121,187,167),(13,80,188,34),(14,123,189,161),(15,74,190,36),(16,125,191,163),(17,83,63,45),(18,134,64,172),(19,85,57,47),(20,136,58,174),(21,87,59,41),(22,130,60,176),(23,81,61,43),(24,132,62,170),(25,91,71,53),(26,142,72,180),(27,93,65,55),(28,144,66,182),(29,95,67,49),(30,138,68,184),(31,89,69,51),(32,140,70,178),(33,97,79,105),(35,99,73,107),(37,101,75,109),(39,103,77,111),(50,118,96,156),(52,120,90,158),(54,114,92,160),(56,116,94,154),(98,168,106,122),(100,162,108,124),(102,164,110,126),(104,166,112,128),(113,179,159,141),(115,181,153,143),(117,183,155,137),(119,177,157,139)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12X | 24A | ··· | 24P | 24Q | ··· | 24X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | D4 | M4(2) | SD16 | Q16 | C3×D4 | C4≀C2 | C3×M4(2) | C3×SD16 | C3×Q16 | C3×C4≀C2 |
| kernel | C3×Q8⋊C8 | C4×C24 | C3×C4⋊C8 | Q8×C12 | Q8⋊C8 | C3×C4⋊C4 | C6×Q8 | C4×C8 | C4⋊C8 | C4×Q8 | C3×Q8 | C4⋊C4 | C2×Q8 | Q8 | C2×C12 | C12 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of C3×Q8⋊C8 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 1 | 71 |
| 0 | 1 | 72 |
| 1 | 0 | 0 |
| 0 | 20 | 47 |
| 0 | 7 | 53 |
| 22 | 0 | 0 |
| 0 | 63 | 72 |
| 0 | 26 | 10 |
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[1,0,0,0,1,1,0,71,72],[1,0,0,0,20,7,0,47,53],[22,0,0,0,63,26,0,72,10] >;
C3×Q8⋊C8 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes C_8
% in TeX
G:=Group("C3xQ8:C8"); // GroupNames label
G:=SmallGroup(192,132);
// by ID
G=gap.SmallGroup(192,132);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,1683,850,136,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=d^8=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations