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## G = C2×C4×C3⋊C8order 192 = 26·3

### Direct product of C2×C4 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×C3⋊C8
G = < a,b,c,d | a2=b4=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 216 in 162 conjugacy classes, 135 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C42, C2×C8, C22×C4, C22×C4, C3⋊C8, C2×C12, C2×C12, C22×C6, C4×C8, C2×C42, C22×C8, C2×C3⋊C8, C4×C12, C22×C12, C22×C12, C2×C4×C8, C4×C3⋊C8, C22×C3⋊C8, C2×C4×C12, C2×C4×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C42, C2×C8, C22×C4, C3⋊C8, C4×S3, C2×Dic3, C22×S3, C4×C8, C2×C42, C22×C8, C2×C3⋊C8, C4×Dic3, S3×C2×C4, C22×Dic3, C2×C4×C8, C4×C3⋊C8, C22×C3⋊C8, C2×C4×Dic3, C2×C4×C3⋊C8

Smallest permutation representation of C2×C4×C3⋊C8
Regular action on 192 points
Generators in S192
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 108)(42 109)(43 110)(44 111)(45 112)(46 105)(47 106)(48 107)(65 191)(66 192)(67 185)(68 186)(69 187)(70 188)(71 189)(72 190)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 137)(81 148)(82 149)(83 150)(84 151)(85 152)(86 145)(87 146)(88 147)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(113 173)(114 174)(115 175)(116 176)(117 169)(118 170)(119 171)(120 172)(121 162)(122 163)(123 164)(124 165)(125 166)(126 167)(127 168)(128 161)(129 182)(130 183)(131 184)(132 177)(133 178)(134 179)(135 180)(136 181)
(1 16 82 97)(2 9 83 98)(3 10 84 99)(4 11 85 100)(5 12 86 101)(6 13 87 102)(7 14 88 103)(8 15 81 104)(17 74 176 163)(18 75 169 164)(19 76 170 165)(20 77 171 166)(21 78 172 167)(22 79 173 168)(23 80 174 161)(24 73 175 162)(25 141 118 124)(26 142 119 125)(27 143 120 126)(28 144 113 127)(29 137 114 128)(30 138 115 121)(31 139 116 122)(32 140 117 123)(33 54 58 149)(34 55 59 150)(35 56 60 151)(36 49 61 152)(37 50 62 145)(38 51 63 146)(39 52 64 147)(40 53 57 148)(41 155 132 69)(42 156 133 70)(43 157 134 71)(44 158 135 72)(45 159 136 65)(46 160 129 66)(47 153 130 67)(48 154 131 68)(89 183 185 106)(90 184 186 107)(91 177 187 108)(92 178 188 109)(93 179 189 110)(94 180 190 111)(95 181 191 112)(96 182 192 105)
(1 176 110)(2 111 169)(3 170 112)(4 105 171)(5 172 106)(6 107 173)(7 174 108)(8 109 175)(9 94 164)(10 165 95)(11 96 166)(12 167 89)(13 90 168)(14 161 91)(15 92 162)(16 163 93)(17 179 82)(18 83 180)(19 181 84)(20 85 182)(21 183 86)(22 87 184)(23 177 88)(24 81 178)(25 136 151)(26 152 129)(27 130 145)(28 146 131)(29 132 147)(30 148 133)(31 134 149)(32 150 135)(33 139 71)(34 72 140)(35 141 65)(36 66 142)(37 143 67)(38 68 144)(39 137 69)(40 70 138)(41 52 114)(42 115 53)(43 54 116)(44 117 55)(45 56 118)(46 119 49)(47 50 120)(48 113 51)(57 156 121)(58 122 157)(59 158 123)(60 124 159)(61 160 125)(62 126 153)(63 154 127)(64 128 155)(73 104 188)(74 189 97)(75 98 190)(76 191 99)(77 100 192)(78 185 101)(79 102 186)(80 187 103)
(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)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4X 6A ··· 6G 8A ··· 8AF 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + - + image C1 C2 C2 C2 C4 C4 C4 C8 S3 Dic3 D6 Dic3 D6 C3⋊C8 C4×S3 kernel C2×C4×C3⋊C8 C4×C3⋊C8 C22×C3⋊C8 C2×C4×C12 C2×C3⋊C8 C4×C12 C22×C12 C2×C12 C2×C42 C42 C42 C22×C4 C22×C4 C2×C4 C2×C4 # reps 1 4 2 1 16 4 4 32 1 2 2 2 1 16 8

Matrix representation of C2×C4×C3⋊C8 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 46 0 0 0 0 0 27 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 1 72
,
 10 0 0 0 0 0 27 0 0 0 0 0 1 0 0 0 0 0 12 51 0 0 0 63 61

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,72,72],[10,0,0,0,0,0,27,0,0,0,0,0,1,0,0,0,0,0,12,63,0,0,0,51,61] >;

C2×C4×C3⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_3\rtimes C_8
% in TeX

G:=Group("C2xC4xC3:C8");
// GroupNames label

G:=SmallGroup(192,479);
// by ID

G=gap.SmallGroup(192,479);
# by ID

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

G:=Group<a,b,c,d|a^2=b^4=c^3=d^8=1,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