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## G = C2×Dic3⋊C8order 192 = 26·3

### Direct product of C2 and Dic3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×Dic3⋊C8
G = < a,b,c,d | a2=b6=d8=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 248 in 138 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C4⋊C8, C2×C42, C22×C8, C22×C8, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C2×C24, C2×C24, C22×Dic3, C22×C12, C2×C4⋊C8, Dic3⋊C8, C22×C3⋊C8, C2×C4×Dic3, C22×C24, C2×Dic3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, C23, D6, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), S3×C8, C8⋊S3, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C2×C4⋊C8, Dic3⋊C8, S3×C2×C8, C2×C8⋊S3, C2×Dic3⋊C4, C2×Dic3⋊C8

Smallest permutation representation of C2×Dic3⋊C8
Regular action on 192 points
Generators in S192
(1 40)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 113)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(25 96)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(41 136)(42 129)(43 130)(44 131)(45 132)(46 133)(47 134)(48 135)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 145)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 137)(73 184)(74 177)(75 178)(76 179)(77 180)(78 181)(79 182)(80 183)(81 192)(82 185)(83 186)(84 187)(85 188)(86 189)(87 190)(88 191)(97 162)(98 163)(99 164)(100 165)(101 166)(102 167)(103 168)(104 161)(105 127)(106 128)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)(153 170)(154 171)(155 172)(156 173)(157 174)(158 175)(159 176)(160 169)
(1 149 31 51 137 134)(2 150 32 52 138 135)(3 151 25 53 139 136)(4 152 26 54 140 129)(5 145 27 55 141 130)(6 146 28 56 142 131)(7 147 29 49 143 132)(8 148 30 50 144 133)(9 81 158 126 168 79)(10 82 159 127 161 80)(11 83 160 128 162 73)(12 84 153 121 163 74)(13 85 154 122 164 75)(14 86 155 123 165 76)(15 87 156 124 166 77)(16 88 157 125 167 78)(17 67 42 35 63 89)(18 68 43 36 64 90)(19 69 44 37 57 91)(20 70 45 38 58 92)(21 71 46 39 59 93)(22 72 47 40 60 94)(23 65 48 33 61 95)(24 66 41 34 62 96)(97 184 116 186 169 106)(98 177 117 187 170 107)(99 178 118 188 171 108)(100 179 119 189 172 109)(101 180 120 190 173 110)(102 181 113 191 174 111)(103 182 114 192 175 112)(104 183 115 185 176 105)
(1 11 51 128)(2 121 52 12)(3 13 53 122)(4 123 54 14)(5 15 55 124)(6 125 56 16)(7 9 49 126)(8 127 50 10)(17 119 35 109)(18 110 36 120)(19 113 37 111)(20 112 38 114)(21 115 39 105)(22 106 40 116)(23 117 33 107)(24 108 34 118)(25 164 136 85)(26 86 129 165)(27 166 130 87)(28 88 131 167)(29 168 132 81)(30 82 133 161)(31 162 134 83)(32 84 135 163)(41 188 96 99)(42 100 89 189)(43 190 90 101)(44 102 91 191)(45 192 92 103)(46 104 93 185)(47 186 94 97)(48 98 95 187)(57 174 69 181)(58 182 70 175)(59 176 71 183)(60 184 72 169)(61 170 65 177)(62 178 66 171)(63 172 67 179)(64 180 68 173)(73 137 160 149)(74 150 153 138)(75 139 154 151)(76 152 155 140)(77 141 156 145)(78 146 157 142)(79 143 158 147)(80 148 159 144)
(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)| (1,40)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,113)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(25,96)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(41,136)(42,129)(43,130)(44,131)(45,132)(46,133)(47,134)(48,135)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,145)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,137)(73,184)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(81,192)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,161)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(153,170)(154,171)(155,172)(156,173)(157,174)(158,175)(159,176)(160,169), (1,149,31,51,137,134)(2,150,32,52,138,135)(3,151,25,53,139,136)(4,152,26,54,140,129)(5,145,27,55,141,130)(6,146,28,56,142,131)(7,147,29,49,143,132)(8,148,30,50,144,133)(9,81,158,126,168,79)(10,82,159,127,161,80)(11,83,160,128,162,73)(12,84,153,121,163,74)(13,85,154,122,164,75)(14,86,155,123,165,76)(15,87,156,124,166,77)(16,88,157,125,167,78)(17,67,42,35,63,89)(18,68,43,36,64,90)(19,69,44,37,57,91)(20,70,45,38,58,92)(21,71,46,39,59,93)(22,72,47,40,60,94)(23,65,48,33,61,95)(24,66,41,34,62,96)(97,184,116,186,169,106)(98,177,117,187,170,107)(99,178,118,188,171,108)(100,179,119,189,172,109)(101,180,120,190,173,110)(102,181,113,191,174,111)(103,182,114,192,175,112)(104,183,115,185,176,105), (1,11,51,128)(2,121,52,12)(3,13,53,122)(4,123,54,14)(5,15,55,124)(6,125,56,16)(7,9,49,126)(8,127,50,10)(17,119,35,109)(18,110,36,120)(19,113,37,111)(20,112,38,114)(21,115,39,105)(22,106,40,116)(23,117,33,107)(24,108,34,118)(25,164,136,85)(26,86,129,165)(27,166,130,87)(28,88,131,167)(29,168,132,81)(30,82,133,161)(31,162,134,83)(32,84,135,163)(41,188,96,99)(42,100,89,189)(43,190,90,101)(44,102,91,191)(45,192,92,103)(46,104,93,185)(47,186,94,97)(48,98,95,187)(57,174,69,181)(58,182,70,175)(59,176,71,183)(60,184,72,169)(61,170,65,177)(62,178,66,171)(63,172,67,179)(64,180,68,173)(73,137,160,149)(74,150,153,138)(75,139,154,151)(76,152,155,140)(77,141,156,145)(78,146,157,142)(79,143,158,147)(80,148,159,144), (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( (1,40)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,113)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(25,96)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(41,136)(42,129)(43,130)(44,131)(45,132)(46,133)(47,134)(48,135)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,145)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,137)(73,184)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(81,192)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,161)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)(153,170)(154,171)(155,172)(156,173)(157,174)(158,175)(159,176)(160,169), (1,149,31,51,137,134)(2,150,32,52,138,135)(3,151,25,53,139,136)(4,152,26,54,140,129)(5,145,27,55,141,130)(6,146,28,56,142,131)(7,147,29,49,143,132)(8,148,30,50,144,133)(9,81,158,126,168,79)(10,82,159,127,161,80)(11,83,160,128,162,73)(12,84,153,121,163,74)(13,85,154,122,164,75)(14,86,155,123,165,76)(15,87,156,124,166,77)(16,88,157,125,167,78)(17,67,42,35,63,89)(18,68,43,36,64,90)(19,69,44,37,57,91)(20,70,45,38,58,92)(21,71,46,39,59,93)(22,72,47,40,60,94)(23,65,48,33,61,95)(24,66,41,34,62,96)(97,184,116,186,169,106)(98,177,117,187,170,107)(99,178,118,188,171,108)(100,179,119,189,172,109)(101,180,120,190,173,110)(102,181,113,191,174,111)(103,182,114,192,175,112)(104,183,115,185,176,105), (1,11,51,128)(2,121,52,12)(3,13,53,122)(4,123,54,14)(5,15,55,124)(6,125,56,16)(7,9,49,126)(8,127,50,10)(17,119,35,109)(18,110,36,120)(19,113,37,111)(20,112,38,114)(21,115,39,105)(22,106,40,116)(23,117,33,107)(24,108,34,118)(25,164,136,85)(26,86,129,165)(27,166,130,87)(28,88,131,167)(29,168,132,81)(30,82,133,161)(31,162,134,83)(32,84,135,163)(41,188,96,99)(42,100,89,189)(43,190,90,101)(44,102,91,191)(45,192,92,103)(46,104,93,185)(47,186,94,97)(48,98,95,187)(57,174,69,181)(58,182,70,175)(59,176,71,183)(60,184,72,169)(61,170,65,177)(62,178,66,171)(63,172,67,179)(64,180,68,173)(73,137,160,149)(74,150,153,138)(75,139,154,151)(76,152,155,140)(77,141,156,145)(78,146,157,142)(79,143,158,147)(80,148,159,144), (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([[(1,40),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,113),(17,54),(18,55),(19,56),(20,49),(21,50),(22,51),(23,52),(24,53),(25,96),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(41,136),(42,129),(43,130),(44,131),(45,132),(46,133),(47,134),(48,135),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,145),(65,138),(66,139),(67,140),(68,141),(69,142),(70,143),(71,144),(72,137),(73,184),(74,177),(75,178),(76,179),(77,180),(78,181),(79,182),(80,183),(81,192),(82,185),(83,186),(84,187),(85,188),(86,189),(87,190),(88,191),(97,162),(98,163),(99,164),(100,165),(101,166),(102,167),(103,168),(104,161),(105,127),(106,128),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126),(153,170),(154,171),(155,172),(156,173),(157,174),(158,175),(159,176),(160,169)], [(1,149,31,51,137,134),(2,150,32,52,138,135),(3,151,25,53,139,136),(4,152,26,54,140,129),(5,145,27,55,141,130),(6,146,28,56,142,131),(7,147,29,49,143,132),(8,148,30,50,144,133),(9,81,158,126,168,79),(10,82,159,127,161,80),(11,83,160,128,162,73),(12,84,153,121,163,74),(13,85,154,122,164,75),(14,86,155,123,165,76),(15,87,156,124,166,77),(16,88,157,125,167,78),(17,67,42,35,63,89),(18,68,43,36,64,90),(19,69,44,37,57,91),(20,70,45,38,58,92),(21,71,46,39,59,93),(22,72,47,40,60,94),(23,65,48,33,61,95),(24,66,41,34,62,96),(97,184,116,186,169,106),(98,177,117,187,170,107),(99,178,118,188,171,108),(100,179,119,189,172,109),(101,180,120,190,173,110),(102,181,113,191,174,111),(103,182,114,192,175,112),(104,183,115,185,176,105)], [(1,11,51,128),(2,121,52,12),(3,13,53,122),(4,123,54,14),(5,15,55,124),(6,125,56,16),(7,9,49,126),(8,127,50,10),(17,119,35,109),(18,110,36,120),(19,113,37,111),(20,112,38,114),(21,115,39,105),(22,106,40,116),(23,117,33,107),(24,108,34,118),(25,164,136,85),(26,86,129,165),(27,166,130,87),(28,88,131,167),(29,168,132,81),(30,82,133,161),(31,162,134,83),(32,84,135,163),(41,188,96,99),(42,100,89,189),(43,190,90,101),(44,102,91,191),(45,192,92,103),(46,104,93,185),(47,186,94,97),(48,98,95,187),(57,174,69,181),(58,182,70,175),(59,176,71,183),(60,184,72,169),(61,170,65,177),(62,178,66,171),(63,172,67,179),(64,180,68,173),(73,137,160,149),(74,150,153,138),(75,139,154,151),(76,152,155,140),(77,141,156,145),(78,146,157,142),(79,143,158,147),(80,148,159,144)], [(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 ··· 2G 3 4A ··· 4H 4I ··· 4P 6A ··· 6G 8A ··· 8H 8I ··· 8P 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 6 ··· 6 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 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 C8 S3 D4 Q8 D6 D6 M4(2) Dic6 C4×S3 C3⋊D4 C4×S3 S3×C8 C8⋊S3 kernel C2×Dic3⋊C8 Dic3⋊C8 C22×C3⋊C8 C2×C4×Dic3 C22×C24 C4×Dic3 C22×Dic3 C2×Dic3 C22×C8 C2×C12 C2×C12 C2×C8 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 4 4 16 1 2 2 2 1 4 4 2 4 2 8 8

Matrix representation of C2×Dic3⋊C8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 1 72
,
 1 0 0 0 0 0 43 11 0 0 0 11 30 0 0 0 0 0 18 50 0 0 0 68 55
,
 51 0 0 0 0 0 0 72 0 0 0 1 0 0 0 0 0 0 27 0 0 0 0 0 27

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,72,72],[1,0,0,0,0,0,43,11,0,0,0,11,30,0,0,0,0,0,18,68,0,0,0,50,55],[51,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,27] >;

C2×Dic3⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic3:C8");
// GroupNames label

G:=SmallGroup(192,658);
// by ID

G=gap.SmallGroup(192,658);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,58,136,6278]);
// Polycyclic

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

׿
×
𝔽