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## G = C6×C4⋊C8order 192 = 26·3

### Direct product of C6 and C4⋊C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×C4⋊C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C2×C24 — C3×C4⋊C8 — C6×C4⋊C8
 Lower central C1 — C2 — C6×C4⋊C8
 Upper central C1 — C22×C12 — C6×C4⋊C8

Generators and relations for C6×C4⋊C8
G = < a,b,c | a6=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 162 in 138 conjugacy classes, 114 normal (30 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, C42, C2×C8, C2×C8, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4⋊C8, C2×C42, C22×C8, C4×C12, C2×C24, C2×C24, C22×C12, C2×C4⋊C8, C3×C4⋊C8, C2×C4×C12, C22×C24, C6×C4⋊C8
Quotients:

Smallest permutation representation of C6×C4⋊C8
Regular action on 192 points
Generators in S192
(1 83 139 147 19 155)(2 84 140 148 20 156)(3 85 141 149 21 157)(4 86 142 150 22 158)(5 87 143 151 23 159)(6 88 144 152 24 160)(7 81 137 145 17 153)(8 82 138 146 18 154)(9 65 113 125 73 185)(10 66 114 126 74 186)(11 67 115 127 75 187)(12 68 116 128 76 188)(13 69 117 121 77 189)(14 70 118 122 78 190)(15 71 119 123 79 191)(16 72 120 124 80 192)(25 89 37 49 41 161)(26 90 38 50 42 162)(27 91 39 51 43 163)(28 92 40 52 44 164)(29 93 33 53 45 165)(30 94 34 54 46 166)(31 95 35 55 47 167)(32 96 36 56 48 168)(57 103 169 112 177 129)(58 104 170 105 178 130)(59 97 171 106 179 131)(60 98 172 107 180 132)(61 99 173 108 181 133)(62 100 174 109 182 134)(63 101 175 110 183 135)(64 102 176 111 184 136)
(1 171 95 79)(2 80 96 172)(3 173 89 73)(4 74 90 174)(5 175 91 75)(6 76 92 176)(7 169 93 77)(8 78 94 170)(9 141 181 49)(10 50 182 142)(11 143 183 51)(12 52 184 144)(13 137 177 53)(14 54 178 138)(15 139 179 55)(16 56 180 140)(17 57 165 117)(18 118 166 58)(19 59 167 119)(20 120 168 60)(21 61 161 113)(22 114 162 62)(23 63 163 115)(24 116 164 64)(25 125 157 99)(26 100 158 126)(27 127 159 101)(28 102 160 128)(29 121 153 103)(30 104 154 122)(31 123 155 97)(32 98 156 124)(33 189 81 112)(34 105 82 190)(35 191 83 106)(36 107 84 192)(37 185 85 108)(38 109 86 186)(39 187 87 110)(40 111 88 188)(41 65 149 133)(42 134 150 66)(43 67 151 135)(44 136 152 68)(45 69 145 129)(46 130 146 70)(47 71 147 131)(48 132 148 72)
(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)| (1,83,139,147,19,155)(2,84,140,148,20,156)(3,85,141,149,21,157)(4,86,142,150,22,158)(5,87,143,151,23,159)(6,88,144,152,24,160)(7,81,137,145,17,153)(8,82,138,146,18,154)(9,65,113,125,73,185)(10,66,114,126,74,186)(11,67,115,127,75,187)(12,68,116,128,76,188)(13,69,117,121,77,189)(14,70,118,122,78,190)(15,71,119,123,79,191)(16,72,120,124,80,192)(25,89,37,49,41,161)(26,90,38,50,42,162)(27,91,39,51,43,163)(28,92,40,52,44,164)(29,93,33,53,45,165)(30,94,34,54,46,166)(31,95,35,55,47,167)(32,96,36,56,48,168)(57,103,169,112,177,129)(58,104,170,105,178,130)(59,97,171,106,179,131)(60,98,172,107,180,132)(61,99,173,108,181,133)(62,100,174,109,182,134)(63,101,175,110,183,135)(64,102,176,111,184,136), (1,171,95,79)(2,80,96,172)(3,173,89,73)(4,74,90,174)(5,175,91,75)(6,76,92,176)(7,169,93,77)(8,78,94,170)(9,141,181,49)(10,50,182,142)(11,143,183,51)(12,52,184,144)(13,137,177,53)(14,54,178,138)(15,139,179,55)(16,56,180,140)(17,57,165,117)(18,118,166,58)(19,59,167,119)(20,120,168,60)(21,61,161,113)(22,114,162,62)(23,63,163,115)(24,116,164,64)(25,125,157,99)(26,100,158,126)(27,127,159,101)(28,102,160,128)(29,121,153,103)(30,104,154,122)(31,123,155,97)(32,98,156,124)(33,189,81,112)(34,105,82,190)(35,191,83,106)(36,107,84,192)(37,185,85,108)(38,109,86,186)(39,187,87,110)(40,111,88,188)(41,65,149,133)(42,134,150,66)(43,67,151,135)(44,136,152,68)(45,69,145,129)(46,130,146,70)(47,71,147,131)(48,132,148,72), (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( (1,83,139,147,19,155)(2,84,140,148,20,156)(3,85,141,149,21,157)(4,86,142,150,22,158)(5,87,143,151,23,159)(6,88,144,152,24,160)(7,81,137,145,17,153)(8,82,138,146,18,154)(9,65,113,125,73,185)(10,66,114,126,74,186)(11,67,115,127,75,187)(12,68,116,128,76,188)(13,69,117,121,77,189)(14,70,118,122,78,190)(15,71,119,123,79,191)(16,72,120,124,80,192)(25,89,37,49,41,161)(26,90,38,50,42,162)(27,91,39,51,43,163)(28,92,40,52,44,164)(29,93,33,53,45,165)(30,94,34,54,46,166)(31,95,35,55,47,167)(32,96,36,56,48,168)(57,103,169,112,177,129)(58,104,170,105,178,130)(59,97,171,106,179,131)(60,98,172,107,180,132)(61,99,173,108,181,133)(62,100,174,109,182,134)(63,101,175,110,183,135)(64,102,176,111,184,136), (1,171,95,79)(2,80,96,172)(3,173,89,73)(4,74,90,174)(5,175,91,75)(6,76,92,176)(7,169,93,77)(8,78,94,170)(9,141,181,49)(10,50,182,142)(11,143,183,51)(12,52,184,144)(13,137,177,53)(14,54,178,138)(15,139,179,55)(16,56,180,140)(17,57,165,117)(18,118,166,58)(19,59,167,119)(20,120,168,60)(21,61,161,113)(22,114,162,62)(23,63,163,115)(24,116,164,64)(25,125,157,99)(26,100,158,126)(27,127,159,101)(28,102,160,128)(29,121,153,103)(30,104,154,122)(31,123,155,97)(32,98,156,124)(33,189,81,112)(34,105,82,190)(35,191,83,106)(36,107,84,192)(37,185,85,108)(38,109,86,186)(39,187,87,110)(40,111,88,188)(41,65,149,133)(42,134,150,66)(43,67,151,135)(44,136,152,68)(45,69,145,129)(46,130,146,70)(47,71,147,131)(48,132,148,72), (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([[(1,83,139,147,19,155),(2,84,140,148,20,156),(3,85,141,149,21,157),(4,86,142,150,22,158),(5,87,143,151,23,159),(6,88,144,152,24,160),(7,81,137,145,17,153),(8,82,138,146,18,154),(9,65,113,125,73,185),(10,66,114,126,74,186),(11,67,115,127,75,187),(12,68,116,128,76,188),(13,69,117,121,77,189),(14,70,118,122,78,190),(15,71,119,123,79,191),(16,72,120,124,80,192),(25,89,37,49,41,161),(26,90,38,50,42,162),(27,91,39,51,43,163),(28,92,40,52,44,164),(29,93,33,53,45,165),(30,94,34,54,46,166),(31,95,35,55,47,167),(32,96,36,56,48,168),(57,103,169,112,177,129),(58,104,170,105,178,130),(59,97,171,106,179,131),(60,98,172,107,180,132),(61,99,173,108,181,133),(62,100,174,109,182,134),(63,101,175,110,183,135),(64,102,176,111,184,136)], [(1,171,95,79),(2,80,96,172),(3,173,89,73),(4,74,90,174),(5,175,91,75),(6,76,92,176),(7,169,93,77),(8,78,94,170),(9,141,181,49),(10,50,182,142),(11,143,183,51),(12,52,184,144),(13,137,177,53),(14,54,178,138),(15,139,179,55),(16,56,180,140),(17,57,165,117),(18,118,166,58),(19,59,167,119),(20,120,168,60),(21,61,161,113),(22,114,162,62),(23,63,163,115),(24,116,164,64),(25,125,157,99),(26,100,158,126),(27,127,159,101),(28,102,160,128),(29,121,153,103),(30,104,154,122),(31,123,155,97),(32,98,156,124),(33,189,81,112),(34,105,82,190),(35,191,83,106),(36,107,84,192),(37,185,85,108),(38,109,86,186),(39,187,87,110),(40,111,88,188),(41,65,149,133),(42,134,150,66),(43,67,151,135),(44,136,152,68),(45,69,145,129),(46,130,146,70),(47,71,147,131),(48,132,148,72)], [(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)]])

120 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 4I ··· 4P 6A ··· 6N 8A ··· 8P 12A ··· 12P 12Q ··· 12AF 24A ··· 24AF order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 ··· 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 D4 Q8 M4(2) C3×D4 C3×Q8 C3×M4(2) kernel C6×C4⋊C8 C3×C4⋊C8 C2×C4×C12 C22×C24 C2×C4⋊C8 C4×C12 C22×C12 C4⋊C8 C2×C42 C22×C8 C2×C12 C42 C22×C4 C2×C4 C2×C12 C2×C12 C2×C6 C2×C4 C2×C4 C22 # reps 1 4 1 2 2 4 4 8 2 4 16 8 8 32 2 2 4 4 4 8

Matrix representation of C6×C4⋊C8 in GL4(𝔽73) generated by

 65 0 0 0 0 72 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 10 0 0 0 0 46 0 0 0 0 41 46 0 0 46 32
G:=sub<GL(4,GF(73))| [65,0,0,0,0,72,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[10,0,0,0,0,46,0,0,0,0,41,46,0,0,46,32] >;

C6×C4⋊C8 in GAP, Magma, Sage, TeX

C_6\times C_4\rtimes C_8
% in TeX

G:=Group("C6xC4:C8");
// GroupNames label

G:=SmallGroup(192,855);
// by ID

G=gap.SmallGroup(192,855);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,124]);
// Polycyclic

G:=Group<a,b,c|a^6=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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