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## G = C6×C2.D8order 192 = 26·3

### Direct product of C6 and C2.D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C2.D8 — C6×C2.D8
 Lower central C1 — C2 — C4 — C6×C2.D8
 Upper central C1 — C22×C6 — C22×C12 — C6×C2.D8

Generators and relations for C6×C2.D8
G = < a,b,c,d | a6=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 194 in 130 conjugacy classes, 98 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C2.D8, C2×C4⋊C4, C22×C8, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C22×C12, C2×C2.D8, C3×C2.D8, C6×C4⋊C4, C22×C24, C6×C2.D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C3×C4⋊C4, C3×D8, C3×Q16, C22×C12, C6×D4, C6×Q8, C2×C2.D8, C3×C2.D8, C6×C4⋊C4, C6×D8, C6×Q16, C6×C2.D8

Smallest permutation representation of C6×C2.D8
Regular action on 192 points
Generators in S192
(1 15 50 71 25 87)(2 16 51 72 26 88)(3 9 52 65 27 81)(4 10 53 66 28 82)(5 11 54 67 29 83)(6 12 55 68 30 84)(7 13 56 69 31 85)(8 14 49 70 32 86)(17 63 125 37 89 48)(18 64 126 38 90 41)(19 57 127 39 91 42)(20 58 128 40 92 43)(21 59 121 33 93 44)(22 60 122 34 94 45)(23 61 123 35 95 46)(24 62 124 36 96 47)(73 104 188 165 120 142)(74 97 189 166 113 143)(75 98 190 167 114 144)(76 99 191 168 115 137)(77 100 192 161 116 138)(78 101 185 162 117 139)(79 102 186 163 118 140)(80 103 187 164 119 141)(105 132 173 152 181 157)(106 133 174 145 182 158)(107 134 175 146 183 159)(108 135 176 147 184 160)(109 136 169 148 177 153)(110 129 170 149 178 154)(111 130 171 150 179 155)(112 131 172 151 180 156)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 89)(32 90)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 86)(42 87)(43 88)(44 81)(45 82)(46 83)(47 84)(48 85)(49 126)(50 127)(51 128)(52 121)(53 122)(54 123)(55 124)(56 125)(73 135)(74 136)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(97 169)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(105 139)(106 140)(107 141)(108 142)(109 143)(110 144)(111 137)(112 138)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)(145 186)(146 187)(147 188)(148 189)(149 190)(150 191)(151 192)(152 185)(161 180)(162 181)(163 182)(164 183)(165 184)(166 177)(167 178)(168 179)
(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)
(1 172 19 100)(2 171 20 99)(3 170 21 98)(4 169 22 97)(5 176 23 104)(6 175 24 103)(7 174 17 102)(8 173 18 101)(9 149 59 190)(10 148 60 189)(11 147 61 188)(12 146 62 187)(13 145 63 186)(14 152 64 185)(15 151 57 192)(16 150 58 191)(25 112 91 138)(26 111 92 137)(27 110 93 144)(28 109 94 143)(29 108 95 142)(30 107 96 141)(31 106 89 140)(32 105 90 139)(33 114 65 154)(34 113 66 153)(35 120 67 160)(36 119 68 159)(37 118 69 158)(38 117 70 157)(39 116 71 156)(40 115 72 155)(41 78 86 132)(42 77 87 131)(43 76 88 130)(44 75 81 129)(45 74 82 136)(46 73 83 135)(47 80 84 134)(48 79 85 133)(49 181 126 162)(50 180 127 161)(51 179 128 168)(52 178 121 167)(53 177 122 166)(54 184 123 165)(55 183 124 164)(56 182 125 163)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 Q8 D4 D8 Q16 C3×D4 C3×Q8 C3×D4 C3×D8 C3×Q16 kernel C6×C2.D8 C3×C2.D8 C6×C4⋊C4 C22×C24 C2×C2.D8 C2×C24 C2.D8 C2×C4⋊C4 C22×C8 C2×C8 C2×C12 C2×C12 C22×C6 C2×C6 C2×C6 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 2 8 8 4 2 16 1 2 1 4 4 2 4 2 8 8

Matrix representation of C6×C2.D8 in GL4(𝔽73) generated by

 9 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 0 1 0 0 0 0 57 16 0 0 57 57
,
 72 0 0 0 0 46 0 0 0 0 50 28 0 0 28 23
G:=sub<GL(4,GF(73))| [9,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,1,0,0,0,0,57,57,0,0,16,57],[72,0,0,0,0,46,0,0,0,0,50,28,0,0,28,23] >;

C6×C2.D8 in GAP, Magma, Sage, TeX

C_6\times C_2.D_8
% in TeX

G:=Group("C6xC2.D8");
// GroupNames label

G:=SmallGroup(192,859);
// by ID

G=gap.SmallGroup(192,859);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,848,4204,172]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=b,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

׿
×
𝔽