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## G = C6×C4.Q8order 192 = 26·3

### Direct product of C6 and C4.Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×C4.Q8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C4.Q8 — C6×C4.Q8
 Lower central C1 — C2 — C4 — C6×C4.Q8
 Upper central C1 — C22×C6 — C22×C12 — C6×C4.Q8

Generators and relations for C6×C4.Q8
G = < a,b,c,d | a6=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 194 in 130 conjugacy classes, 98 normal (22 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, C4.Q8, C2×C4⋊C4, C22×C8, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C22×C12, C2×C4.Q8, C3×C4.Q8, C6×C4⋊C4, C22×C24, C6×C4.Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C4.Q8, C2×C4⋊C4, C2×SD16, C3×C4⋊C4, C3×SD16, C22×C12, C6×D4, C6×Q8, C2×C4.Q8, C3×C4.Q8, C6×C4⋊C4, C6×SD16, C6×C4.Q8

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

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

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

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

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 type + + + + + - + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 Q8 D4 SD16 C3×D4 C3×Q8 C3×D4 C3×SD16 kernel C6×C4.Q8 C3×C4.Q8 C6×C4⋊C4 C22×C24 C2×C4.Q8 C2×C24 C4.Q8 C2×C4⋊C4 C22×C8 C2×C8 C2×C12 C2×C12 C22×C6 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 2 8 8 4 2 16 1 2 1 8 2 4 2 16

Matrix representation of C6×C4.Q8 in GL5(𝔽73)

 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 71 0 0 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 0 0 61 61 0 0 0 6 0
,
 1 0 0 0 0 0 67 67 0 0 0 67 6 0 0 0 0 0 59 64 0 0 0 46 14

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,1,0,0,0,71,1],[1,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,61,6,0,0,0,61,0],[1,0,0,0,0,0,67,67,0,0,0,67,6,0,0,0,0,0,59,46,0,0,0,64,14] >;

C6×C4.Q8 in GAP, Magma, Sage, TeX

C_6\times C_4.Q_8
% in TeX

G:=Group("C6xC4.Q8");
// GroupNames label

G:=SmallGroup(192,858);
// by ID

G=gap.SmallGroup(192,858);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,4204,172]);
// Polycyclic

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

׿
×
𝔽