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## G = C2×C8⋊Dic3order 192 = 26·3

### Direct product of C2 and C8⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C8⋊Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — C2×C8⋊Dic3
 Lower central C3 — C6 — C12 — C2×C8⋊Dic3
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×C8⋊Dic3
G = < a,b,c,d | a2=b8=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >

Subgroups: 312 in 130 conjugacy classes, 87 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4.Q8, C2×C4⋊C4, C22×C8, C4⋊Dic3, C4⋊Dic3, C2×C24, C22×Dic3, C22×C12, C2×C4.Q8, C8⋊Dic3, C2×C4⋊Dic3, C22×C24, C2×C8⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C4.Q8, C2×C4⋊C4, C2×SD16, C24⋊C2, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C2×C4.Q8, C8⋊Dic3, C2×C24⋊C2, C2×C4⋊Dic3, C2×C8⋊Dic3

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + + - + + image C1 C2 C2 C2 C4 S3 D4 Q8 D4 Dic3 D6 D6 SD16 Dic6 D12 D12 C24⋊C2 kernel C2×C8⋊Dic3 C8⋊Dic3 C2×C4⋊Dic3 C22×C24 C2×C24 C22×C8 C2×C12 C2×C12 C22×C6 C2×C8 C2×C8 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 8 1 1 2 1 4 2 1 8 4 2 2 16

Matrix representation of C2×C8⋊Dic3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 67 6 0 0 0 0 67 67 0 0 0 0 0 0 67 6 0 0 0 0 67 67 0 0 0 0 0 0 48 62 0 0 0 0 11 37
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 1
,
 71 56 0 0 0 0 56 2 0 0 0 0 0 0 57 53 0 0 0 0 53 16 0 0 0 0 0 0 68 19 0 0 0 0 14 5

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,0,0,48,11,0,0,0,0,62,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[71,56,0,0,0,0,56,2,0,0,0,0,0,0,57,53,0,0,0,0,53,16,0,0,0,0,0,0,68,14,0,0,0,0,19,5] >;

C2×C8⋊Dic3 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC8:Dic3");
// GroupNames label

G:=SmallGroup(192,663);
// by ID

G=gap.SmallGroup(192,663);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,100,1684,102,6278]);
// 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,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

׿
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