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## G = C2×C42.S3order 192 = 26·3

### Direct product of C2 and C42.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C42.S3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C2×C42.S3
 Lower central C3 — C6 — C2×C42.S3
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C42.S3
G = < a,b,c,d | a2=b6=d4=1, c4=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 216 in 146 conjugacy classes, 103 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C2×C8, C22×C4, C22×C4, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C8⋊C4, C2×C42, C22×C8, C2×C3⋊C8, C4×C12, C22×C12, C22×C12, C2×C8⋊C4, C42.S3, C22×C3⋊C8, C2×C4×C12, C2×C42.S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, M4(2), C22×C4, C4×S3, C2×Dic3, C22×S3, C8⋊C4, C2×C42, C2×M4(2), C4.Dic3, C4×Dic3, S3×C2×C4, C22×Dic3, C2×C8⋊C4, C42.S3, C2×C4.Dic3, C2×C4×Dic3, C2×C42.S3

Smallest permutation representation of C2×C42.S3
Regular action on 192 points
Generators in S192
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(25 128)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 176)(34 169)(35 170)(36 171)(37 172)(38 173)(39 174)(40 175)(41 188)(42 189)(43 190)(44 191)(45 192)(46 185)(47 186)(48 187)(49 134)(50 135)(51 136)(52 129)(53 130)(54 131)(55 132)(56 133)(65 103)(66 104)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 152)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 165)(82 166)(83 167)(84 168)(85 161)(86 162)(87 163)(88 164)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 153)(105 140)(106 141)(107 142)(108 143)(109 144)(110 137)(111 138)(112 139)(113 178)(114 179)(115 180)(116 181)(117 182)(118 183)(119 184)(120 177)
(1 50 26 5 54 30)(2 31 55 6 27 51)(3 52 28 7 56 32)(4 25 49 8 29 53)(9 114 142 13 118 138)(10 139 119 14 143 115)(11 116 144 15 120 140)(12 141 113 16 137 117)(17 124 130 21 128 134)(18 135 121 22 131 125)(19 126 132 23 122 136)(20 129 123 24 133 127)(33 97 163 37 101 167)(34 168 102 38 164 98)(35 99 165 39 103 161)(36 162 104 40 166 100)(41 94 145 45 90 149)(42 150 91 46 146 95)(43 96 147 47 92 151)(44 152 93 48 148 89)(57 183 111 61 179 107)(58 108 180 62 112 184)(59 177 105 63 181 109)(60 110 182 64 106 178)(65 85 170 69 81 174)(66 175 82 70 171 86)(67 87 172 71 83 176)(68 169 84 72 173 88)(73 158 187 77 154 191)(74 192 155 78 188 159)(75 160 189 79 156 185)(76 186 157 80 190 153)
(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)
(1 154 179 67)(2 159 180 72)(3 156 181 69)(4 153 182 66)(5 158 183 71)(6 155 184 68)(7 160 177 65)(8 157 178 70)(9 33 125 148)(10 38 126 145)(11 35 127 150)(12 40 128 147)(13 37 121 152)(14 34 122 149)(15 39 123 146)(16 36 124 151)(17 92 113 100)(18 89 114 97)(19 94 115 102)(20 91 116 99)(21 96 117 104)(22 93 118 101)(23 90 119 98)(24 95 120 103)(25 76 64 175)(26 73 57 172)(27 78 58 169)(28 75 59 174)(29 80 60 171)(30 77 61 176)(31 74 62 173)(32 79 63 170)(41 143 168 136)(42 140 161 133)(43 137 162 130)(44 142 163 135)(45 139 164 132)(46 144 165 129)(47 141 166 134)(48 138 167 131)(49 186 106 82)(50 191 107 87)(51 188 108 84)(52 185 109 81)(53 190 110 86)(54 187 111 83)(55 192 112 88)(56 189 105 85)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4P 6A ··· 6G 8A ··· 8P 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + - + image C1 C2 C2 C2 C4 C4 C4 S3 Dic3 D6 Dic3 D6 M4(2) C4×S3 C4.Dic3 kernel C2×C42.S3 C42.S3 C22×C3⋊C8 C2×C4×C12 C2×C3⋊C8 C4×C12 C22×C12 C2×C42 C42 C42 C22×C4 C22×C4 C2×C6 C2×C4 C22 # reps 1 4 2 1 16 4 4 1 2 2 2 1 8 8 16

Matrix representation of C2×C42.S3 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 65 0 0 0 16 9
,
 1 0 0 0 0 72 0 0 0 0 19 43 0 0 16 54
,
 27 0 0 0 0 1 0 0 0 0 27 0 0 0 5 46
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,65,16,0,0,0,9],[1,0,0,0,0,72,0,0,0,0,19,16,0,0,43,54],[27,0,0,0,0,1,0,0,0,0,27,5,0,0,0,46] >;

C2×C42.S3 in GAP, Magma, Sage, TeX

C_2\times C_4^2.S_3
% in TeX

G:=Group("C2xC4^2.S3");
// GroupNames label

G:=SmallGroup(192,480);
// by ID

G=gap.SmallGroup(192,480);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,100,136,6278]);
// Polycyclic

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

׿
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𝔽