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## G = C3×Q64order 192 = 26·3

### Direct product of C3 and Q64

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C3×Q64, C32.C6, Q32.C6, C96.2C2, C24.66D4, C6.17D16, C12.41D8, C48.21C22, C8.7(C3×D4), C4.3(C3×D8), C16.4(C2×C6), C2.5(C3×D16), (C3×Q32).2C2, SmallGroup(192,179)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C16 — C3×Q64
 Chief series C1 — C2 — C4 — C8 — C16 — C48 — C3×Q32 — C3×Q64
 Lower central C1 — C2 — C4 — C8 — C16 — C3×Q64
 Upper central C1 — C6 — C12 — C24 — C48 — C3×Q64

Generators and relations for C3×Q64
G = < a,b,c | a3=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Q64
Regular action on 192 points
Generators in S192
(1 141 106)(2 142 107)(3 143 108)(4 144 109)(5 145 110)(6 146 111)(7 147 112)(8 148 113)(9 149 114)(10 150 115)(11 151 116)(12 152 117)(13 153 118)(14 154 119)(15 155 120)(16 156 121)(17 157 122)(18 158 123)(19 159 124)(20 160 125)(21 129 126)(22 130 127)(23 131 128)(24 132 97)(25 133 98)(26 134 99)(27 135 100)(28 136 101)(29 137 102)(30 138 103)(31 139 104)(32 140 105)(33 76 164)(34 77 165)(35 78 166)(36 79 167)(37 80 168)(38 81 169)(39 82 170)(40 83 171)(41 84 172)(42 85 173)(43 86 174)(44 87 175)(45 88 176)(46 89 177)(47 90 178)(48 91 179)(49 92 180)(50 93 181)(51 94 182)(52 95 183)(53 96 184)(54 65 185)(55 66 186)(56 67 187)(57 68 188)(58 69 189)(59 70 190)(60 71 191)(61 72 192)(62 73 161)(63 74 162)(64 75 163)
(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 85 17 69)(2 84 18 68)(3 83 19 67)(4 82 20 66)(5 81 21 65)(6 80 22 96)(7 79 23 95)(8 78 24 94)(9 77 25 93)(10 76 26 92)(11 75 27 91)(12 74 28 90)(13 73 29 89)(14 72 30 88)(15 71 31 87)(16 70 32 86)(33 99 49 115)(34 98 50 114)(35 97 51 113)(36 128 52 112)(37 127 53 111)(38 126 54 110)(39 125 55 109)(40 124 56 108)(41 123 57 107)(42 122 58 106)(43 121 59 105)(44 120 60 104)(45 119 61 103)(46 118 62 102)(47 117 63 101)(48 116 64 100)(129 185 145 169)(130 184 146 168)(131 183 147 167)(132 182 148 166)(133 181 149 165)(134 180 150 164)(135 179 151 163)(136 178 152 162)(137 177 153 161)(138 176 154 192)(139 175 155 191)(140 174 156 190)(141 173 157 189)(142 172 158 188)(143 171 159 187)(144 170 160 186)

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

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

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

57 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 6A 6B 8A 8B 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D 24A 24B 24C 24D 32A ··· 32H 48A ··· 48H 96A ··· 96P order 1 2 3 3 4 4 4 6 6 8 8 12 12 12 12 12 12 16 16 16 16 24 24 24 24 32 ··· 32 48 ··· 48 96 ··· 96 size 1 1 1 1 2 16 16 1 1 2 2 2 2 16 16 16 16 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C3 C6 C6 D4 D8 C3×D4 D16 C3×D8 Q64 C3×D16 C3×Q64 kernel C3×Q64 C96 C3×Q32 Q64 C32 Q32 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 1 2 2 4 4 8 8 16

Matrix representation of C3×Q64 in GL2(𝔽31) generated by

 25 0 0 25
,
 0 30 1 21
,
 8 18 5 23
G:=sub<GL(2,GF(31))| [25,0,0,25],[0,1,30,21],[8,5,18,23] >;

C3×Q64 in GAP, Magma, Sage, TeX

C_3\times Q_{64}
% in TeX

G:=Group("C3xQ64");
// GroupNames label

G:=SmallGroup(192,179);
// by ID

G=gap.SmallGroup(192,179);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,672,197,680,1011,514,192,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c|a^3=b^32=1,c^2=b^16,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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