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## G = C3×C8⋊Q8order 192 = 26·3

### Direct product of C3 and C8⋊Q8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8⋊Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C4⋊Q8 — C3×C8⋊Q8
 Lower central C1 — C2 — C2×C4 — C3×C8⋊Q8
 Upper central C1 — C2×C6 — C4×C12 — C3×C8⋊Q8

Generators and relations for C3×C8⋊Q8
G = < a,b,c,d | a3=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=c-1 >

Subgroups: 146 in 90 conjugacy classes, 58 normal (30 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, C8⋊Q8, C3×C8⋊C4, C3×C4.Q8, C3×C2.D8, C3×C42.C2, C3×C4⋊Q8, C3×C8⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C3×D4, C3×Q8, C22×C6, C4⋊Q8, C8⋊C22, C8.C22, C6×D4, C6×Q8, C8⋊Q8, C3×C4⋊Q8, C3×C8⋊C22, C3×C8.C22, C3×C8⋊Q8

Smallest permutation representation of C3×C8⋊Q8
Regular action on 192 points
Generators in S192
(1 37 16)(2 38 9)(3 39 10)(4 40 11)(5 33 12)(6 34 13)(7 35 14)(8 36 15)(17 56 41)(18 49 42)(19 50 43)(20 51 44)(21 52 45)(22 53 46)(23 54 47)(24 55 48)(25 189 162)(26 190 163)(27 191 164)(28 192 165)(29 185 166)(30 186 167)(31 187 168)(32 188 161)(57 77 82)(58 78 83)(59 79 84)(60 80 85)(61 73 86)(62 74 87)(63 75 88)(64 76 81)(65 104 90)(66 97 91)(67 98 92)(68 99 93)(69 100 94)(70 101 95)(71 102 96)(72 103 89)(105 130 127)(106 131 128)(107 132 121)(108 133 122)(109 134 123)(110 135 124)(111 136 125)(112 129 126)(113 152 139)(114 145 140)(115 146 141)(116 147 142)(117 148 143)(118 149 144)(119 150 137)(120 151 138)(153 173 180)(154 174 181)(155 175 182)(156 176 183)(157 169 184)(158 170 177)(159 171 178)(160 172 179)
(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 66 23 62)(2 71 24 59)(3 68 17 64)(4 65 18 61)(5 70 19 58)(6 67 20 63)(7 72 21 60)(8 69 22 57)(9 96 48 84)(10 93 41 81)(11 90 42 86)(12 95 43 83)(13 92 44 88)(14 89 45 85)(15 94 46 82)(16 91 47 87)(25 136 172 150)(26 133 173 147)(27 130 174 152)(28 135 175 149)(29 132 176 146)(30 129 169 151)(31 134 170 148)(32 131 171 145)(33 101 50 78)(34 98 51 75)(35 103 52 80)(36 100 53 77)(37 97 54 74)(38 102 55 79)(39 99 56 76)(40 104 49 73)(105 154 113 164)(106 159 114 161)(107 156 115 166)(108 153 116 163)(109 158 117 168)(110 155 118 165)(111 160 119 162)(112 157 120 167)(121 183 141 185)(122 180 142 190)(123 177 143 187)(124 182 144 192)(125 179 137 189)(126 184 138 186)(127 181 139 191)(128 178 140 188)
(1 115 23 107)(2 118 24 110)(3 113 17 105)(4 116 18 108)(5 119 19 111)(6 114 20 106)(7 117 21 109)(8 120 22 112)(9 144 48 124)(10 139 41 127)(11 142 42 122)(12 137 43 125)(13 140 44 128)(14 143 45 123)(15 138 46 126)(16 141 47 121)(25 101 172 78)(26 104 173 73)(27 99 174 76)(28 102 175 79)(29 97 176 74)(30 100 169 77)(31 103 170 80)(32 98 171 75)(33 150 50 136)(34 145 51 131)(35 148 52 134)(36 151 53 129)(37 146 54 132)(38 149 55 135)(39 152 56 130)(40 147 49 133)(57 167 69 157)(58 162 70 160)(59 165 71 155)(60 168 72 158)(61 163 65 153)(62 166 66 156)(63 161 67 159)(64 164 68 154)(81 191 93 181)(82 186 94 184)(83 189 95 179)(84 192 96 182)(85 187 89 177)(86 190 90 180)(87 185 91 183)(88 188 92 178)

G:=sub<Sym(192)| (1,37,16)(2,38,9)(3,39,10)(4,40,11)(5,33,12)(6,34,13)(7,35,14)(8,36,15)(17,56,41)(18,49,42)(19,50,43)(20,51,44)(21,52,45)(22,53,46)(23,54,47)(24,55,48)(25,189,162)(26,190,163)(27,191,164)(28,192,165)(29,185,166)(30,186,167)(31,187,168)(32,188,161)(57,77,82)(58,78,83)(59,79,84)(60,80,85)(61,73,86)(62,74,87)(63,75,88)(64,76,81)(65,104,90)(66,97,91)(67,98,92)(68,99,93)(69,100,94)(70,101,95)(71,102,96)(72,103,89)(105,130,127)(106,131,128)(107,132,121)(108,133,122)(109,134,123)(110,135,124)(111,136,125)(112,129,126)(113,152,139)(114,145,140)(115,146,141)(116,147,142)(117,148,143)(118,149,144)(119,150,137)(120,151,138)(153,173,180)(154,174,181)(155,175,182)(156,176,183)(157,169,184)(158,170,177)(159,171,178)(160,172,179), (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), (1,66,23,62)(2,71,24,59)(3,68,17,64)(4,65,18,61)(5,70,19,58)(6,67,20,63)(7,72,21,60)(8,69,22,57)(9,96,48,84)(10,93,41,81)(11,90,42,86)(12,95,43,83)(13,92,44,88)(14,89,45,85)(15,94,46,82)(16,91,47,87)(25,136,172,150)(26,133,173,147)(27,130,174,152)(28,135,175,149)(29,132,176,146)(30,129,169,151)(31,134,170,148)(32,131,171,145)(33,101,50,78)(34,98,51,75)(35,103,52,80)(36,100,53,77)(37,97,54,74)(38,102,55,79)(39,99,56,76)(40,104,49,73)(105,154,113,164)(106,159,114,161)(107,156,115,166)(108,153,116,163)(109,158,117,168)(110,155,118,165)(111,160,119,162)(112,157,120,167)(121,183,141,185)(122,180,142,190)(123,177,143,187)(124,182,144,192)(125,179,137,189)(126,184,138,186)(127,181,139,191)(128,178,140,188), (1,115,23,107)(2,118,24,110)(3,113,17,105)(4,116,18,108)(5,119,19,111)(6,114,20,106)(7,117,21,109)(8,120,22,112)(9,144,48,124)(10,139,41,127)(11,142,42,122)(12,137,43,125)(13,140,44,128)(14,143,45,123)(15,138,46,126)(16,141,47,121)(25,101,172,78)(26,104,173,73)(27,99,174,76)(28,102,175,79)(29,97,176,74)(30,100,169,77)(31,103,170,80)(32,98,171,75)(33,150,50,136)(34,145,51,131)(35,148,52,134)(36,151,53,129)(37,146,54,132)(38,149,55,135)(39,152,56,130)(40,147,49,133)(57,167,69,157)(58,162,70,160)(59,165,71,155)(60,168,72,158)(61,163,65,153)(62,166,66,156)(63,161,67,159)(64,164,68,154)(81,191,93,181)(82,186,94,184)(83,189,95,179)(84,192,96,182)(85,187,89,177)(86,190,90,180)(87,185,91,183)(88,188,92,178)>;

G:=Group( (1,37,16)(2,38,9)(3,39,10)(4,40,11)(5,33,12)(6,34,13)(7,35,14)(8,36,15)(17,56,41)(18,49,42)(19,50,43)(20,51,44)(21,52,45)(22,53,46)(23,54,47)(24,55,48)(25,189,162)(26,190,163)(27,191,164)(28,192,165)(29,185,166)(30,186,167)(31,187,168)(32,188,161)(57,77,82)(58,78,83)(59,79,84)(60,80,85)(61,73,86)(62,74,87)(63,75,88)(64,76,81)(65,104,90)(66,97,91)(67,98,92)(68,99,93)(69,100,94)(70,101,95)(71,102,96)(72,103,89)(105,130,127)(106,131,128)(107,132,121)(108,133,122)(109,134,123)(110,135,124)(111,136,125)(112,129,126)(113,152,139)(114,145,140)(115,146,141)(116,147,142)(117,148,143)(118,149,144)(119,150,137)(120,151,138)(153,173,180)(154,174,181)(155,175,182)(156,176,183)(157,169,184)(158,170,177)(159,171,178)(160,172,179), (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), (1,66,23,62)(2,71,24,59)(3,68,17,64)(4,65,18,61)(5,70,19,58)(6,67,20,63)(7,72,21,60)(8,69,22,57)(9,96,48,84)(10,93,41,81)(11,90,42,86)(12,95,43,83)(13,92,44,88)(14,89,45,85)(15,94,46,82)(16,91,47,87)(25,136,172,150)(26,133,173,147)(27,130,174,152)(28,135,175,149)(29,132,176,146)(30,129,169,151)(31,134,170,148)(32,131,171,145)(33,101,50,78)(34,98,51,75)(35,103,52,80)(36,100,53,77)(37,97,54,74)(38,102,55,79)(39,99,56,76)(40,104,49,73)(105,154,113,164)(106,159,114,161)(107,156,115,166)(108,153,116,163)(109,158,117,168)(110,155,118,165)(111,160,119,162)(112,157,120,167)(121,183,141,185)(122,180,142,190)(123,177,143,187)(124,182,144,192)(125,179,137,189)(126,184,138,186)(127,181,139,191)(128,178,140,188), (1,115,23,107)(2,118,24,110)(3,113,17,105)(4,116,18,108)(5,119,19,111)(6,114,20,106)(7,117,21,109)(8,120,22,112)(9,144,48,124)(10,139,41,127)(11,142,42,122)(12,137,43,125)(13,140,44,128)(14,143,45,123)(15,138,46,126)(16,141,47,121)(25,101,172,78)(26,104,173,73)(27,99,174,76)(28,102,175,79)(29,97,176,74)(30,100,169,77)(31,103,170,80)(32,98,171,75)(33,150,50,136)(34,145,51,131)(35,148,52,134)(36,151,53,129)(37,146,54,132)(38,149,55,135)(39,152,56,130)(40,147,49,133)(57,167,69,157)(58,162,70,160)(59,165,71,155)(60,168,72,158)(61,163,65,153)(62,166,66,156)(63,161,67,159)(64,164,68,154)(81,191,93,181)(82,186,94,184)(83,189,95,179)(84,192,96,182)(85,187,89,177)(86,190,90,180)(87,185,91,183)(88,188,92,178) );

G=PermutationGroup([[(1,37,16),(2,38,9),(3,39,10),(4,40,11),(5,33,12),(6,34,13),(7,35,14),(8,36,15),(17,56,41),(18,49,42),(19,50,43),(20,51,44),(21,52,45),(22,53,46),(23,54,47),(24,55,48),(25,189,162),(26,190,163),(27,191,164),(28,192,165),(29,185,166),(30,186,167),(31,187,168),(32,188,161),(57,77,82),(58,78,83),(59,79,84),(60,80,85),(61,73,86),(62,74,87),(63,75,88),(64,76,81),(65,104,90),(66,97,91),(67,98,92),(68,99,93),(69,100,94),(70,101,95),(71,102,96),(72,103,89),(105,130,127),(106,131,128),(107,132,121),(108,133,122),(109,134,123),(110,135,124),(111,136,125),(112,129,126),(113,152,139),(114,145,140),(115,146,141),(116,147,142),(117,148,143),(118,149,144),(119,150,137),(120,151,138),(153,173,180),(154,174,181),(155,175,182),(156,176,183),(157,169,184),(158,170,177),(159,171,178),(160,172,179)], [(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)], [(1,66,23,62),(2,71,24,59),(3,68,17,64),(4,65,18,61),(5,70,19,58),(6,67,20,63),(7,72,21,60),(8,69,22,57),(9,96,48,84),(10,93,41,81),(11,90,42,86),(12,95,43,83),(13,92,44,88),(14,89,45,85),(15,94,46,82),(16,91,47,87),(25,136,172,150),(26,133,173,147),(27,130,174,152),(28,135,175,149),(29,132,176,146),(30,129,169,151),(31,134,170,148),(32,131,171,145),(33,101,50,78),(34,98,51,75),(35,103,52,80),(36,100,53,77),(37,97,54,74),(38,102,55,79),(39,99,56,76),(40,104,49,73),(105,154,113,164),(106,159,114,161),(107,156,115,166),(108,153,116,163),(109,158,117,168),(110,155,118,165),(111,160,119,162),(112,157,120,167),(121,183,141,185),(122,180,142,190),(123,177,143,187),(124,182,144,192),(125,179,137,189),(126,184,138,186),(127,181,139,191),(128,178,140,188)], [(1,115,23,107),(2,118,24,110),(3,113,17,105),(4,116,18,108),(5,119,19,111),(6,114,20,106),(7,117,21,109),(8,120,22,112),(9,144,48,124),(10,139,41,127),(11,142,42,122),(12,137,43,125),(13,140,44,128),(14,143,45,123),(15,138,46,126),(16,141,47,121),(25,101,172,78),(26,104,173,73),(27,99,174,76),(28,102,175,79),(29,97,176,74),(30,100,169,77),(31,103,170,80),(32,98,171,75),(33,150,50,136),(34,145,51,131),(35,148,52,134),(36,151,53,129),(37,146,54,132),(38,149,55,135),(39,152,56,130),(40,147,49,133),(57,167,69,157),(58,162,70,160),(59,165,71,155),(60,168,72,158),(61,163,65,153),(62,166,66,156),(63,161,67,159),(64,164,68,154),(81,191,93,181),(82,186,94,184),(83,189,95,179),(84,192,96,182),(85,187,89,177),(86,190,90,180),(87,185,91,183),(88,188,92,178)]])

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 24A ··· 24H 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 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 2 2 4 4 8 8 8 8 1 ··· 1 4 4 4 4 2 2 2 2 4 4 4 4 8 ··· 8 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 Q8 D4 C3×Q8 C3×D4 C8⋊C22 C8.C22 C3×C8⋊C22 C3×C8.C22 kernel C3×C8⋊Q8 C3×C8⋊C4 C3×C4.Q8 C3×C2.D8 C3×C42.C2 C3×C4⋊Q8 C8⋊Q8 C8⋊C4 C4.Q8 C2.D8 C42.C2 C4⋊Q8 C24 C2×C12 C8 C2×C4 C6 C6 C2 C2 # reps 1 1 2 2 1 1 2 2 4 4 2 2 4 2 8 4 1 1 2 2

Matrix representation of C3×C8⋊Q8 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 72 0 0 0
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 28 34 12 49 0 0 39 28 24 12 0 0 49 61 45 39 0 0 12 49 34 45
,
 62 43 0 0 0 0 43 11 0 0 0 0 0 0 20 50 45 61 0 0 50 53 61 28 0 0 28 12 23 20 0 0 12 45 20 50

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,28,39,49,12,0,0,34,28,61,49,0,0,12,24,45,34,0,0,49,12,39,45],[62,43,0,0,0,0,43,11,0,0,0,0,0,0,20,50,28,12,0,0,50,53,12,45,0,0,45,61,23,20,0,0,61,28,20,50] >;

C3×C8⋊Q8 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes Q_8
% in TeX

G:=Group("C3xC8:Q8");
// GroupNames label

G:=SmallGroup(192,934);
// by ID

G=gap.SmallGroup(192,934);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,176,1094,1059,268,6053,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

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