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## G = C3×C8⋊1C8order 192 = 26·3

### Direct product of C3 and C8⋊1C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C8⋊1C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C12 — C3×C4⋊C8 — C3×C8⋊1C8
 Lower central C1 — C2 — C4 — C3×C8⋊1C8
 Upper central C1 — C2×C12 — C4×C12 — C3×C8⋊1C8

Generators and relations for C3×C81C8
G = < a,b,c | a3=b8=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×C81C8
Regular action on 192 points
Generators in S192
(1 76 17)(2 77 18)(3 78 19)(4 79 20)(5 80 21)(6 73 22)(7 74 23)(8 75 24)(9 135 58)(10 136 59)(11 129 60)(12 130 61)(13 131 62)(14 132 63)(15 133 64)(16 134 57)(25 72 88)(26 65 81)(27 66 82)(28 67 83)(29 68 84)(30 69 85)(31 70 86)(32 71 87)(33 105 92)(34 106 93)(35 107 94)(36 108 95)(37 109 96)(38 110 89)(39 111 90)(40 112 91)(41 97 146)(42 98 147)(43 99 148)(44 100 149)(45 101 150)(46 102 151)(47 103 152)(48 104 145)(49 189 138)(50 190 139)(51 191 140)(52 192 141)(53 185 142)(54 186 143)(55 187 144)(56 188 137)(113 163 155)(114 164 156)(115 165 157)(116 166 158)(117 167 159)(118 168 160)(119 161 153)(120 162 154)(121 183 172)(122 184 173)(123 177 174)(124 178 175)(125 179 176)(126 180 169)(127 181 170)(128 182 171)
(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)
(1 140 72 156 34 132 42 128)(2 139 65 155 35 131 43 127)(3 138 66 154 36 130 44 126)(4 137 67 153 37 129 45 125)(5 144 68 160 38 136 46 124)(6 143 69 159 39 135 47 123)(7 142 70 158 40 134 48 122)(8 141 71 157 33 133 41 121)(9 152 174 22 186 30 167 90)(10 151 175 21 187 29 168 89)(11 150 176 20 188 28 161 96)(12 149 169 19 189 27 162 95)(13 148 170 18 190 26 163 94)(14 147 171 17 191 25 164 93)(15 146 172 24 192 32 165 92)(16 145 173 23 185 31 166 91)(49 82 120 108 61 100 180 78)(50 81 113 107 62 99 181 77)(51 88 114 106 63 98 182 76)(52 87 115 105 64 97 183 75)(53 86 116 112 57 104 184 74)(54 85 117 111 58 103 177 73)(55 84 118 110 59 102 178 80)(56 83 119 109 60 101 179 79)

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

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

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

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 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C6 C6 C8 C12 C24 D4 Q8 M4(2) D8 Q16 C3×D4 C3×Q8 C8.C4 C3×M4(2) C3×D8 C3×Q16 C3×C8.C4 kernel C3×C8⋊1C8 C4×C24 C3×C4⋊C8 C8⋊1C8 C2×C24 C4×C8 C4⋊C8 C24 C2×C8 C8 C2×C12 C2×C12 C12 C12 C12 C2×C4 C2×C4 C6 C4 C4 C4 C2 # reps 1 1 2 2 4 2 4 8 8 16 1 1 2 2 2 2 2 4 4 4 4 8

Matrix representation of C3×C81C8 in GL3(𝔽73) generated by

 64 0 0 0 64 0 0 0 64
,
 1 0 0 0 16 57 0 16 16
,
 22 0 0 0 26 64 0 64 47
G:=sub<GL(3,GF(73))| [64,0,0,0,64,0,0,0,64],[1,0,0,0,16,16,0,57,16],[22,0,0,0,26,64,0,64,47] >;

C3×C81C8 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_1C_8
% in TeX

G:=Group("C3xC8:1C8");
// GroupNames label

G:=SmallGroup(192,141);
// by ID

G=gap.SmallGroup(192,141);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,428,1683,136,172]);
// 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^-1>;
// generators/relations

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