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## G = Dic3⋊C16order 192 = 26·3

### The semidirect product of Dic3 and C16 acting via C16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Dic3⋊C16
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C8×Dic3 — Dic3⋊C16
 Lower central C3 — C6 — Dic3⋊C16
 Upper central C1 — C2×C8 — C2×C16

Generators and relations for Dic3⋊C16
G = < a,b,c | a6=c16=1, b2=a3, bab-1=a-1, ac=ca, cbc-1=a3b >

Smallest permutation representation of Dic3⋊C16
Regular action on 192 points
Generators in S192
(1 42 110 140 116 188)(2 43 111 141 117 189)(3 44 112 142 118 190)(4 45 97 143 119 191)(5 46 98 144 120 192)(6 47 99 129 121 177)(7 48 100 130 122 178)(8 33 101 131 123 179)(9 34 102 132 124 180)(10 35 103 133 125 181)(11 36 104 134 126 182)(12 37 105 135 127 183)(13 38 106 136 128 184)(14 39 107 137 113 185)(15 40 108 138 114 186)(16 41 109 139 115 187)(17 59 175 89 158 73)(18 60 176 90 159 74)(19 61 161 91 160 75)(20 62 162 92 145 76)(21 63 163 93 146 77)(22 64 164 94 147 78)(23 49 165 95 148 79)(24 50 166 96 149 80)(25 51 167 81 150 65)(26 52 168 82 151 66)(27 53 169 83 152 67)(28 54 170 84 153 68)(29 55 171 85 154 69)(30 56 172 86 155 70)(31 57 173 87 156 71)(32 58 174 88 157 72)
(1 17 140 89)(2 90 141 18)(3 19 142 91)(4 92 143 20)(5 21 144 93)(6 94 129 22)(7 23 130 95)(8 96 131 24)(9 25 132 81)(10 82 133 26)(11 27 134 83)(12 84 135 28)(13 29 136 85)(14 86 137 30)(15 31 138 87)(16 88 139 32)(33 166 123 80)(34 65 124 167)(35 168 125 66)(36 67 126 169)(37 170 127 68)(38 69 128 171)(39 172 113 70)(40 71 114 173)(41 174 115 72)(42 73 116 175)(43 176 117 74)(44 75 118 161)(45 162 119 76)(46 77 120 163)(47 164 121 78)(48 79 122 165)(49 100 148 178)(50 179 149 101)(51 102 150 180)(52 181 151 103)(53 104 152 182)(54 183 153 105)(55 106 154 184)(56 185 155 107)(57 108 156 186)(58 187 157 109)(59 110 158 188)(60 189 159 111)(61 112 160 190)(62 191 145 97)(63 98 146 192)(64 177 147 99)
(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)| (1,42,110,140,116,188)(2,43,111,141,117,189)(3,44,112,142,118,190)(4,45,97,143,119,191)(5,46,98,144,120,192)(6,47,99,129,121,177)(7,48,100,130,122,178)(8,33,101,131,123,179)(9,34,102,132,124,180)(10,35,103,133,125,181)(11,36,104,134,126,182)(12,37,105,135,127,183)(13,38,106,136,128,184)(14,39,107,137,113,185)(15,40,108,138,114,186)(16,41,109,139,115,187)(17,59,175,89,158,73)(18,60,176,90,159,74)(19,61,161,91,160,75)(20,62,162,92,145,76)(21,63,163,93,146,77)(22,64,164,94,147,78)(23,49,165,95,148,79)(24,50,166,96,149,80)(25,51,167,81,150,65)(26,52,168,82,151,66)(27,53,169,83,152,67)(28,54,170,84,153,68)(29,55,171,85,154,69)(30,56,172,86,155,70)(31,57,173,87,156,71)(32,58,174,88,157,72), (1,17,140,89)(2,90,141,18)(3,19,142,91)(4,92,143,20)(5,21,144,93)(6,94,129,22)(7,23,130,95)(8,96,131,24)(9,25,132,81)(10,82,133,26)(11,27,134,83)(12,84,135,28)(13,29,136,85)(14,86,137,30)(15,31,138,87)(16,88,139,32)(33,166,123,80)(34,65,124,167)(35,168,125,66)(36,67,126,169)(37,170,127,68)(38,69,128,171)(39,172,113,70)(40,71,114,173)(41,174,115,72)(42,73,116,175)(43,176,117,74)(44,75,118,161)(45,162,119,76)(46,77,120,163)(47,164,121,78)(48,79,122,165)(49,100,148,178)(50,179,149,101)(51,102,150,180)(52,181,151,103)(53,104,152,182)(54,183,153,105)(55,106,154,184)(56,185,155,107)(57,108,156,186)(58,187,157,109)(59,110,158,188)(60,189,159,111)(61,112,160,190)(62,191,145,97)(63,98,146,192)(64,177,147,99), (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( (1,42,110,140,116,188)(2,43,111,141,117,189)(3,44,112,142,118,190)(4,45,97,143,119,191)(5,46,98,144,120,192)(6,47,99,129,121,177)(7,48,100,130,122,178)(8,33,101,131,123,179)(9,34,102,132,124,180)(10,35,103,133,125,181)(11,36,104,134,126,182)(12,37,105,135,127,183)(13,38,106,136,128,184)(14,39,107,137,113,185)(15,40,108,138,114,186)(16,41,109,139,115,187)(17,59,175,89,158,73)(18,60,176,90,159,74)(19,61,161,91,160,75)(20,62,162,92,145,76)(21,63,163,93,146,77)(22,64,164,94,147,78)(23,49,165,95,148,79)(24,50,166,96,149,80)(25,51,167,81,150,65)(26,52,168,82,151,66)(27,53,169,83,152,67)(28,54,170,84,153,68)(29,55,171,85,154,69)(30,56,172,86,155,70)(31,57,173,87,156,71)(32,58,174,88,157,72), (1,17,140,89)(2,90,141,18)(3,19,142,91)(4,92,143,20)(5,21,144,93)(6,94,129,22)(7,23,130,95)(8,96,131,24)(9,25,132,81)(10,82,133,26)(11,27,134,83)(12,84,135,28)(13,29,136,85)(14,86,137,30)(15,31,138,87)(16,88,139,32)(33,166,123,80)(34,65,124,167)(35,168,125,66)(36,67,126,169)(37,170,127,68)(38,69,128,171)(39,172,113,70)(40,71,114,173)(41,174,115,72)(42,73,116,175)(43,176,117,74)(44,75,118,161)(45,162,119,76)(46,77,120,163)(47,164,121,78)(48,79,122,165)(49,100,148,178)(50,179,149,101)(51,102,150,180)(52,181,151,103)(53,104,152,182)(54,183,153,105)(55,106,154,184)(56,185,155,107)(57,108,156,186)(58,187,157,109)(59,110,158,188)(60,189,159,111)(61,112,160,190)(62,191,145,97)(63,98,146,192)(64,177,147,99), (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([[(1,42,110,140,116,188),(2,43,111,141,117,189),(3,44,112,142,118,190),(4,45,97,143,119,191),(5,46,98,144,120,192),(6,47,99,129,121,177),(7,48,100,130,122,178),(8,33,101,131,123,179),(9,34,102,132,124,180),(10,35,103,133,125,181),(11,36,104,134,126,182),(12,37,105,135,127,183),(13,38,106,136,128,184),(14,39,107,137,113,185),(15,40,108,138,114,186),(16,41,109,139,115,187),(17,59,175,89,158,73),(18,60,176,90,159,74),(19,61,161,91,160,75),(20,62,162,92,145,76),(21,63,163,93,146,77),(22,64,164,94,147,78),(23,49,165,95,148,79),(24,50,166,96,149,80),(25,51,167,81,150,65),(26,52,168,82,151,66),(27,53,169,83,152,67),(28,54,170,84,153,68),(29,55,171,85,154,69),(30,56,172,86,155,70),(31,57,173,87,156,71),(32,58,174,88,157,72)], [(1,17,140,89),(2,90,141,18),(3,19,142,91),(4,92,143,20),(5,21,144,93),(6,94,129,22),(7,23,130,95),(8,96,131,24),(9,25,132,81),(10,82,133,26),(11,27,134,83),(12,84,135,28),(13,29,136,85),(14,86,137,30),(15,31,138,87),(16,88,139,32),(33,166,123,80),(34,65,124,167),(35,168,125,66),(36,67,126,169),(37,170,127,68),(38,69,128,171),(39,172,113,70),(40,71,114,173),(41,174,115,72),(42,73,116,175),(43,176,117,74),(44,75,118,161),(45,162,119,76),(46,77,120,163),(47,164,121,78),(48,79,122,165),(49,100,148,178),(50,179,149,101),(51,102,150,180),(52,181,151,103),(53,104,152,182),(54,183,153,105),(55,106,154,184),(56,185,155,107),(57,108,156,186),(58,187,157,109),(59,110,158,188),(60,189,159,111),(61,112,160,190),(62,191,145,97),(63,98,146,192),(64,177,147,99)], [(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)]])

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A ··· 8H 8I 8J 8K 8L 12A 12B 12C 12D 16A ··· 16H 16I ··· 16P 24A ··· 24H 48A ··· 48P order 1 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 8 ··· 8 8 8 8 8 12 12 12 12 16 ··· 16 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 2 1 1 1 1 6 6 6 6 2 2 2 1 ··· 1 6 6 6 6 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 C16 S3 D4 Q8 D6 M4(2) Dic6 C3⋊D4 C4×S3 M5(2) C8⋊S3 S3×C8 S3×C16 D6.C8 kernel Dic3⋊C16 C2×C3⋊C16 C8×Dic3 C2×C48 C2×C3⋊C8 C4×Dic3 C2×Dic3 Dic3 C2×C16 C24 C24 C2×C8 C12 C8 C8 C2×C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 2 2 8 16 1 1 1 1 2 2 2 2 4 4 4 8 8

Matrix representation of Dic3⋊C16 in GL3(𝔽97) generated by

 1 0 0 0 1 96 0 1 0
,
 96 0 0 0 80 50 0 33 17
,
 85 0 0 0 92 10 0 87 5
G:=sub<GL(3,GF(97))| [1,0,0,0,1,1,0,96,0],[96,0,0,0,80,33,0,50,17],[85,0,0,0,92,87,0,10,5] >;

Dic3⋊C16 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes C_{16}
% in TeX

G:=Group("Dic3:C16");
// GroupNames label

G:=SmallGroup(192,60);
// by ID

G=gap.SmallGroup(192,60);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,100,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^6=c^16=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations

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