Copied to
clipboard

## G = C2×C6×Q16order 192 = 26·3

### Direct product of C2×C6 and Q16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C2×C6×Q16
 Chief series C1 — C2 — C4 — C12 — C3×Q8 — C3×Q16 — C6×Q16 — C2×C6×Q16
 Lower central C1 — C2 — C4 — C2×C6×Q16
 Upper central C1 — C22×C6 — C22×C12 — C2×C6×Q16

Generators and relations for C2×C6×Q16
G = < a,b,c,d | a2=b6=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 338 in 258 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×C6, C22×C8, C2×Q16, C22×Q8, C2×C24, C3×Q16, C22×C12, C22×C12, C6×Q8, C6×Q8, C22×Q16, C22×C24, C6×Q16, Q8×C2×C6, C2×C6×Q16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, Q16, C2×D4, C24, C3×D4, C22×C6, C2×Q16, C22×D4, C3×Q16, C6×D4, C23×C6, C22×Q16, C6×Q16, D4×C2×C6, C2×C6×Q16

Smallest permutation representation of C2×C6×Q16
Regular action on 192 points
Generators in S192
(1 191)(2 192)(3 185)(4 186)(5 187)(6 188)(7 189)(8 190)(9 133)(10 134)(11 135)(12 136)(13 129)(14 130)(15 131)(16 132)(17 84)(18 85)(19 86)(20 87)(21 88)(22 81)(23 82)(24 83)(25 125)(26 126)(27 127)(28 128)(29 121)(30 122)(31 123)(32 124)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 70)(42 71)(43 72)(44 65)(45 66)(46 67)(47 68)(48 69)(57 170)(58 171)(59 172)(60 173)(61 174)(62 175)(63 176)(64 169)(73 166)(74 167)(75 168)(76 161)(77 162)(78 163)(79 164)(80 165)(89 158)(90 159)(91 160)(92 153)(93 154)(94 155)(95 156)(96 157)(97 178)(98 179)(99 180)(100 181)(101 182)(102 183)(103 184)(104 177)(105 150)(106 151)(107 152)(108 145)(109 146)(110 147)(111 148)(112 149)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 137)
(1 68 109 35 171 179)(2 69 110 36 172 180)(3 70 111 37 173 181)(4 71 112 38 174 182)(5 72 105 39 175 183)(6 65 106 40 176 184)(7 66 107 33 169 177)(8 67 108 34 170 178)(9 137 153 80 88 121)(10 138 154 73 81 122)(11 139 155 74 82 123)(12 140 156 75 83 124)(13 141 157 76 84 125)(14 142 158 77 85 126)(15 143 159 78 86 127)(16 144 160 79 87 128)(17 25 129 116 96 161)(18 26 130 117 89 162)(19 27 131 118 90 163)(20 28 132 119 91 164)(21 29 133 120 92 165)(22 30 134 113 93 166)(23 31 135 114 94 167)(24 32 136 115 95 168)(41 148 51 60 100 185)(42 149 52 61 101 186)(43 150 53 62 102 187)(44 151 54 63 103 188)(45 152 55 64 104 189)(46 145 56 57 97 190)(47 146 49 58 98 191)(48 147 50 59 99 192)
(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)
(1 135 5 131)(2 134 6 130)(3 133 7 129)(4 132 8 136)(9 189 13 185)(10 188 14 192)(11 187 15 191)(12 186 16 190)(17 173 21 169)(18 172 22 176)(19 171 23 175)(20 170 24 174)(25 181 29 177)(26 180 30 184)(27 179 31 183)(28 178 32 182)(33 161 37 165)(34 168 38 164)(35 167 39 163)(36 166 40 162)(41 137 45 141)(42 144 46 140)(43 143 47 139)(44 142 48 138)(49 74 53 78)(50 73 54 77)(51 80 55 76)(52 79 56 75)(57 83 61 87)(58 82 62 86)(59 81 63 85)(60 88 64 84)(65 117 69 113)(66 116 70 120)(67 115 71 119)(68 114 72 118)(89 110 93 106)(90 109 94 105)(91 108 95 112)(92 107 96 111)(97 124 101 128)(98 123 102 127)(99 122 103 126)(100 121 104 125)(145 156 149 160)(146 155 150 159)(147 154 151 158)(148 153 152 157)

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

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

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

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 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 Q16 C3×D4 C3×D4 C3×Q16 kernel C2×C6×Q16 C22×C24 C6×Q16 Q8×C2×C6 C22×Q16 C22×C8 C2×Q16 C22×Q8 C2×C12 C22×C6 C2×C6 C2×C4 C23 C22 # reps 1 1 12 2 2 2 24 4 3 1 8 6 2 16

Matrix representation of C2×C6×Q16 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 65 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 16 57 0 0 16 16
,
 72 0 0 0 0 72 0 0 0 0 67 6 0 0 6 6
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[65,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,16,16,0,0,57,16],[72,0,0,0,0,72,0,0,0,0,67,6,0,0,6,6] >;

C2×C6×Q16 in GAP, Magma, Sage, TeX

C_2\times C_6\times Q_{16}
% in TeX

G:=Group("C2xC6xQ16");
// GroupNames label

G:=SmallGroup(192,1460);
// by ID

G=gap.SmallGroup(192,1460);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,680,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=1,d^2=c^4,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