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## G = C3×C16⋊4C4order 192 = 26·3

### Direct product of C3 and C16⋊4C4

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×C16⋊4C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C24 — C3×C2.D8 — C3×C16⋊4C4
 Lower central C1 — C2 — C4 — C8 — C3×C16⋊4C4
 Upper central C1 — C2×C6 — C2×C12 — C2×C24 — C3×C16⋊4C4

Generators and relations for C3×C164C4
G = < a,b,c | a3=b16=c4=1, ab=ba, ac=ca, cbc-1=b7 >

Smallest permutation representation of C3×C164C4
Regular action on 192 points
Generators in S192
(1 113 160)(2 114 145)(3 115 146)(4 116 147)(5 117 148)(6 118 149)(7 119 150)(8 120 151)(9 121 152)(10 122 153)(11 123 154)(12 124 155)(13 125 156)(14 126 157)(15 127 158)(16 128 159)(17 57 71)(18 58 72)(19 59 73)(20 60 74)(21 61 75)(22 62 76)(23 63 77)(24 64 78)(25 49 79)(26 50 80)(27 51 65)(28 52 66)(29 53 67)(30 54 68)(31 55 69)(32 56 70)(33 136 110)(34 137 111)(35 138 112)(36 139 97)(37 140 98)(38 141 99)(39 142 100)(40 143 101)(41 144 102)(42 129 103)(43 130 104)(44 131 105)(45 132 106)(46 133 107)(47 134 108)(48 135 109)(81 171 184)(82 172 185)(83 173 186)(84 174 187)(85 175 188)(86 176 189)(87 161 190)(88 162 191)(89 163 192)(90 164 177)(91 165 178)(92 166 179)(93 167 180)(94 168 181)(95 169 182)(96 170 183)
(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)
(1 174 99 25)(2 165 100 32)(3 172 101 23)(4 163 102 30)(5 170 103 21)(6 161 104 28)(7 168 105 19)(8 175 106 26)(9 166 107 17)(10 173 108 24)(11 164 109 31)(12 171 110 22)(13 162 111 29)(14 169 112 20)(15 176 97 27)(16 167 98 18)(33 62 124 184)(34 53 125 191)(35 60 126 182)(36 51 127 189)(37 58 128 180)(38 49 113 187)(39 56 114 178)(40 63 115 185)(41 54 116 192)(42 61 117 183)(43 52 118 190)(44 59 119 181)(45 50 120 188)(46 57 121 179)(47 64 122 186)(48 55 123 177)(65 158 86 139)(66 149 87 130)(67 156 88 137)(68 147 89 144)(69 154 90 135)(70 145 91 142)(71 152 92 133)(72 159 93 140)(73 150 94 131)(74 157 95 138)(75 148 96 129)(76 155 81 136)(77 146 82 143)(78 153 83 134)(79 160 84 141)(80 151 85 132)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + - + - + image C1 C2 C2 C3 C4 C6 C6 C12 Q8 D4 Q16 D8 C3×Q8 C3×D4 SD32 C3×Q16 C3×D8 C3×SD32 kernel C3×C16⋊4C4 C3×C2.D8 C2×C48 C16⋊4C4 C48 C2.D8 C2×C16 C16 C24 C2×C12 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 4 4 2 8 1 1 2 2 2 2 8 4 4 16

Matrix representation of C3×C164C4 in GL3(𝔽97) generated by

 35 0 0 0 35 0 0 0 35
,
 96 0 0 0 87 53 0 44 87
,
 22 0 0 0 79 29 0 29 18
G:=sub<GL(3,GF(97))| [35,0,0,0,35,0,0,0,35],[96,0,0,0,87,44,0,53,87],[22,0,0,0,79,29,0,29,18] >;

C3×C164C4 in GAP, Magma, Sage, TeX

C_3\times C_{16}\rtimes_4C_4
% in TeX

G:=Group("C3xC16:4C4");
// GroupNames label

G:=SmallGroup(192,173);
// by ID

G=gap.SmallGroup(192,173);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1100,1683,360,6053,124]);
// Polycyclic

G:=Group<a,b,c|a^3=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations

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