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## G = C2×C6.SD16order 192 = 26·3

### Direct product of C2 and C6.SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C6.SD16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — C22×Dic6 — C2×C6.SD16
 Lower central C3 — C6 — C12 — C2×C6.SD16
 Upper central C1 — C23 — C22×C4 — C2×C4⋊C4

Generators and relations for C2×C6.SD16
G = < a,b,c,d | a2=b6=c8=1, d2=b3c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c-1 >

Subgroups: 376 in 162 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C6 [×3], C6 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×10], C23, Dic3 [×4], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×9], C3⋊C8 [×2], Dic6 [×4], Dic6 [×6], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×4], C22×C6, Q8⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×Q8, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×6], C2×Dic6 [×3], C22×Dic3, C22×C12, C22×C12, C2×Q8⋊C4, C6.SD16 [×4], C22×C3⋊C8, C6×C4⋊C4, C22×Dic6, C2×C6.SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, D6⋊C4 [×4], D4.S3 [×2], C3⋊Q16 [×2], S3×C2×C4, C2×D12, C2×C3⋊D4, C2×Q8⋊C4, C6.SD16 [×4], C2×D6⋊C4, C2×D4.S3, C2×C3⋊Q16, C2×C6.SD16

Smallest permutation representation of C2×C6.SD16
Regular action on 192 points
Generators in S192
(1 118)(2 119)(3 120)(4 113)(5 114)(6 115)(7 116)(8 117)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 146)(18 147)(19 148)(20 149)(21 150)(22 151)(23 152)(24 145)(33 191)(34 192)(35 185)(36 186)(37 187)(38 188)(39 189)(40 190)(41 175)(42 176)(43 169)(44 170)(45 171)(46 172)(47 173)(48 174)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(97 128)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 133)(106 134)(107 135)(108 136)(109 129)(110 130)(111 131)(112 132)(137 165)(138 166)(139 167)(140 168)(141 161)(142 162)(143 163)(144 164)(153 181)(154 182)(155 183)(156 184)(157 177)(158 178)(159 179)(160 180)
(1 109 73 23 189 171)(2 172 190 24 74 110)(3 111 75 17 191 173)(4 174 192 18 76 112)(5 105 77 19 185 175)(6 176 186 20 78 106)(7 107 79 21 187 169)(8 170 188 22 80 108)(9 179 163 103 82 96)(10 89 83 104 164 180)(11 181 165 97 84 90)(12 91 85 98 166 182)(13 183 167 99 86 92)(14 93 87 100 168 184)(15 177 161 101 88 94)(16 95 81 102 162 178)(25 159 143 126 66 56)(26 49 67 127 144 160)(27 153 137 128 68 50)(28 51 69 121 138 154)(29 155 139 122 70 52)(30 53 71 123 140 156)(31 157 141 124 72 54)(32 55 65 125 142 158)(33 47 120 131 59 146)(34 147 60 132 113 48)(35 41 114 133 61 148)(36 149 62 134 115 42)(37 43 116 135 63 150)(38 151 64 136 117 44)(39 45 118 129 57 152)(40 145 58 130 119 46)
(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)
(1 122 19 25)(2 28 20 125)(3 128 21 31)(4 26 22 123)(5 126 23 29)(6 32 24 121)(7 124 17 27)(8 30 18 127)(9 118 99 148)(10 151 100 113)(11 116 101 146)(12 149 102 119)(13 114 103 152)(14 147 104 117)(15 120 97 150)(16 145 98 115)(33 181 135 88)(34 83 136 184)(35 179 129 86)(36 81 130 182)(37 177 131 84)(38 87 132 180)(39 183 133 82)(40 85 134 178)(41 163 57 92)(42 95 58 166)(43 161 59 90)(44 93 60 164)(45 167 61 96)(46 91 62 162)(47 165 63 94)(48 89 64 168)(49 80 140 174)(50 169 141 75)(51 78 142 172)(52 175 143 73)(53 76 144 170)(54 173 137 79)(55 74 138 176)(56 171 139 77)(65 110 154 186)(66 189 155 105)(67 108 156 192)(68 187 157 111)(69 106 158 190)(70 185 159 109)(71 112 160 188)(72 191 153 107)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6G 8A ··· 8H 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 12 12 12 12 2 ··· 2 6 ··· 6 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - + - - image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 SD16 Q16 C4×S3 D12 C3⋊D4 C3⋊D4 D4.S3 C3⋊Q16 kernel C2×C6.SD16 C6.SD16 C22×C3⋊C8 C6×C4⋊C4 C22×Dic6 C2×Dic6 C2×C4⋊C4 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 3 1 2 1 4 4 4 4 2 2 2 2

Matrix representation of C2×C6.SD16 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 5 23 0 0 0 0 18 68 0 0 0 0 0 0 54 14 0 0 0 0 68 19 0 0 0 0 0 0 67 6 0 0 0 0 67 67
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 70 24 0 0 0 0 24 3

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[5,18,0,0,0,0,23,68,0,0,0,0,0,0,54,68,0,0,0,0,14,19,0,0,0,0,0,0,67,67,0,0,0,0,6,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,70,24,0,0,0,0,24,3] >;

C2×C6.SD16 in GAP, Magma, Sage, TeX

C_2\times C_6.{\rm SD}_{16}
% in TeX

G:=Group("C2xC6.SD16");
// GroupNames label

G:=SmallGroup(192,528);
// by ID

G=gap.SmallGroup(192,528);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,422,58,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=1,d^2=b^3*c^4,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^-1>;
// generators/relations

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