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## G = Dic3×C2×C8order 192 = 26·3

### Direct product of C2×C8 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C2×C8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — C2×C4×Dic3 — Dic3×C2×C8
 Lower central C3 — Dic3×C2×C8
 Upper central C1 — C22×C8

Generators and relations for Dic3×C2×C8
G = < a,b,c,d | a2=b8=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 248 in 162 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4×C8, C2×C42, C22×C8, C22×C8, C2×C3⋊C8, C4×Dic3, C2×C24, C22×Dic3, C22×C12, C2×C4×C8, C8×Dic3, C22×C3⋊C8, C2×C4×Dic3, C22×C24, Dic3×C2×C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C42, C2×C8, C22×C4, C4×S3, C2×Dic3, C22×S3, C4×C8, C2×C42, C22×C8, S3×C8, C4×Dic3, S3×C2×C4, C22×Dic3, C2×C4×C8, C8×Dic3, S3×C2×C8, C2×C4×Dic3, Dic3×C2×C8

Smallest permutation representation of Dic3×C2×C8
Regular action on 192 points
Generators in S192
(1 64)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 126)(18 127)(19 128)(20 121)(21 122)(22 123)(23 124)(24 125)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 102)(34 103)(35 104)(36 97)(37 98)(38 99)(39 100)(40 101)(41 138)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 137)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 113)(80 114)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(145 168)(146 161)(147 162)(148 163)(149 164)(150 165)(151 166)(152 167)(153 174)(154 175)(155 176)(156 169)(157 170)(158 171)(159 172)(160 173)(177 185)(178 186)(179 187)(180 188)(181 189)(182 190)(183 191)(184 192)
(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 30 77 24 86 71)(2 31 78 17 87 72)(3 32 79 18 88 65)(4 25 80 19 81 66)(5 26 73 20 82 67)(6 27 74 21 83 68)(7 28 75 22 84 69)(8 29 76 23 85 70)(9 37 47 154 146 192)(10 38 48 155 147 185)(11 39 41 156 148 186)(12 40 42 157 149 187)(13 33 43 158 150 188)(14 34 44 159 151 189)(15 35 45 160 152 190)(16 36 46 153 145 191)(49 57 135 120 126 95)(50 58 136 113 127 96)(51 59 129 114 128 89)(52 60 130 115 121 90)(53 61 131 116 122 91)(54 62 132 117 123 92)(55 63 133 118 124 93)(56 64 134 119 125 94)(97 143 174 168 183 105)(98 144 175 161 184 106)(99 137 176 162 177 107)(100 138 169 163 178 108)(101 139 170 164 179 109)(102 140 171 165 180 110)(103 141 172 166 181 111)(104 142 173 167 182 112)
(1 112 24 173)(2 105 17 174)(3 106 18 175)(4 107 19 176)(5 108 20 169)(6 109 21 170)(7 110 22 171)(8 111 23 172)(9 127 154 58)(10 128 155 59)(11 121 156 60)(12 122 157 61)(13 123 158 62)(14 124 159 63)(15 125 160 64)(16 126 153 57)(25 177 81 137)(26 178 82 138)(27 179 83 139)(28 180 84 140)(29 181 85 141)(30 182 86 142)(31 183 87 143)(32 184 88 144)(33 117 150 54)(34 118 151 55)(35 119 152 56)(36 120 145 49)(37 113 146 50)(38 114 147 51)(39 115 148 52)(40 116 149 53)(41 130 186 90)(42 131 187 91)(43 132 188 92)(44 133 189 93)(45 134 190 94)(46 135 191 95)(47 136 192 96)(48 129 185 89)(65 98 79 161)(66 99 80 162)(67 100 73 163)(68 101 74 164)(69 102 75 165)(70 103 76 166)(71 104 77 167)(72 97 78 168)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4X 6A ··· 6G 8A ··· 8P 8Q ··· 8AF 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 2 1 ··· 1 3 ··· 3 2 ··· 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C8 S3 Dic3 D6 D6 C4×S3 C4×S3 S3×C8 kernel Dic3×C2×C8 C8×Dic3 C22×C3⋊C8 C2×C4×Dic3 C22×C24 C2×C3⋊C8 C4×Dic3 C2×C24 C22×Dic3 C2×Dic3 C22×C8 C2×C8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 4 1 1 1 8 4 8 4 32 1 4 2 1 6 2 16

Matrix representation of Dic3×C2×C8 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 63 0 0 0 0 27 0 0 0 0 46 0 0 0 0 46
,
 72 0 0 0 0 72 0 0 0 0 1 72 0 0 1 0
,
 46 0 0 0 0 27 0 0 0 0 65 47 0 0 39 8
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[63,0,0,0,0,27,0,0,0,0,46,0,0,0,0,46],[72,0,0,0,0,72,0,0,0,0,1,1,0,0,72,0],[46,0,0,0,0,27,0,0,0,0,65,39,0,0,47,8] >;

Dic3×C2×C8 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2\times C_8
% in TeX

G:=Group("Dic3xC2xC8");
// GroupNames label

G:=SmallGroup(192,657);
// by ID

G=gap.SmallGroup(192,657);
# by ID

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

G:=Group<a,b,c,d|a^2=b^8=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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