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## G = C3×C8⋊3Q8order 192 = 26·3

### Direct product of C3 and C8⋊3Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C8⋊3Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C4⋊Q8 — C3×C8⋊3Q8
 Lower central C1 — C2 — C2×C4 — C3×C8⋊3Q8
 Upper central C1 — C2×C6 — C4×C12 — C3×C8⋊3Q8

Generators and relations for C3×C83Q8
G = < a,b,c,d | a3=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >

Subgroups: 162 in 98 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C4×C8, C4.Q8, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, C83Q8, C4×C24, C3×C4.Q8, C3×C4⋊Q8, C3×C83Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, SD16, C2×D4, C2×Q8, C3×D4, C3×Q8, C22×C6, C4⋊Q8, C2×SD16, C3×SD16, C6×D4, C6×Q8, C83Q8, C3×C4⋊Q8, C6×SD16, C3×C83Q8

Smallest permutation representation of C3×C83Q8
Regular action on 192 points
Generators in S192
(1 39 12)(2 40 13)(3 33 14)(4 34 15)(5 35 16)(6 36 9)(7 37 10)(8 38 11)(17 49 41)(18 50 42)(19 51 43)(20 52 44)(21 53 45)(22 54 46)(23 55 47)(24 56 48)(25 189 163)(26 190 164)(27 191 165)(28 192 166)(29 185 167)(30 186 168)(31 187 161)(32 188 162)(57 78 82)(58 79 83)(59 80 84)(60 73 85)(61 74 86)(62 75 87)(63 76 88)(64 77 81)(65 103 89)(66 104 90)(67 97 91)(68 98 92)(69 99 93)(70 100 94)(71 101 95)(72 102 96)(105 130 127)(106 131 128)(107 132 121)(108 133 122)(109 134 123)(110 135 124)(111 136 125)(112 129 126)(113 145 139)(114 146 140)(115 147 141)(116 148 142)(117 149 143)(118 150 144)(119 151 137)(120 152 138)(153 178 175)(154 179 176)(155 180 169)(156 181 170)(157 182 171)(158 183 172)(159 184 173)(160 177 174)
(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 71 23 87)(2 72 24 88)(3 65 17 81)(4 66 18 82)(5 67 19 83)(6 68 20 84)(7 69 21 85)(8 70 22 86)(9 92 44 80)(10 93 45 73)(11 94 46 74)(12 95 47 75)(13 96 48 76)(14 89 41 77)(15 90 42 78)(16 91 43 79)(25 132 180 147)(26 133 181 148)(27 134 182 149)(28 135 183 150)(29 136 184 151)(30 129 177 152)(31 130 178 145)(32 131 179 146)(33 103 49 64)(34 104 50 57)(35 97 51 58)(36 98 52 59)(37 99 53 60)(38 100 54 61)(39 101 55 62)(40 102 56 63)(105 153 113 161)(106 154 114 162)(107 155 115 163)(108 156 116 164)(109 157 117 165)(110 158 118 166)(111 159 119 167)(112 160 120 168)(121 169 141 189)(122 170 142 190)(123 171 143 191)(124 172 144 192)(125 173 137 185)(126 174 138 186)(127 175 139 187)(128 176 140 188)
(1 115 23 107)(2 118 24 110)(3 113 17 105)(4 116 18 108)(5 119 19 111)(6 114 20 106)(7 117 21 109)(8 120 22 112)(9 140 44 128)(10 143 45 123)(11 138 46 126)(12 141 47 121)(13 144 48 124)(14 139 41 127)(15 142 42 122)(16 137 43 125)(25 101 180 62)(26 104 181 57)(27 99 182 60)(28 102 183 63)(29 97 184 58)(30 100 177 61)(31 103 178 64)(32 98 179 59)(33 145 49 130)(34 148 50 133)(35 151 51 136)(36 146 52 131)(37 149 53 134)(38 152 54 129)(39 147 55 132)(40 150 56 135)(65 153 81 161)(66 156 82 164)(67 159 83 167)(68 154 84 162)(69 157 85 165)(70 160 86 168)(71 155 87 163)(72 158 88 166)(73 191 93 171)(74 186 94 174)(75 189 95 169)(76 192 96 172)(77 187 89 175)(78 190 90 170)(79 185 91 173)(80 188 92 176)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4F 4G 4H 4I 4J 6A ··· 6F 8A ··· 8H 12A ··· 12L 12M ··· 12T 24A ··· 24P order 1 2 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 1 2 ··· 2 8 8 8 8 1 ··· 1 2 ··· 2 2 ··· 2 8 ··· 8 2 ··· 2

66 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 Q8 D4 SD16 C3×Q8 C3×D4 C3×SD16 kernel C3×C8⋊3Q8 C4×C24 C3×C4.Q8 C3×C4⋊Q8 C8⋊3Q8 C4×C8 C4.Q8 C4⋊Q8 C24 C2×C12 C12 C8 C2×C4 C4 # reps 1 1 4 2 2 2 8 4 4 2 8 8 4 16

Matrix representation of C3×C83Q8 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 64 0 0 0 0 64
,
 0 61 0 0 6 61 0 0 0 0 67 6 0 0 67 67
,
 72 2 0 0 72 1 0 0 0 0 72 0 0 0 0 72
,
 0 12 0 0 6 0 0 0 0 0 17 71 0 0 71 56
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[0,6,0,0,61,61,0,0,0,0,67,67,0,0,6,67],[72,72,0,0,2,1,0,0,0,0,72,0,0,0,0,72],[0,6,0,0,12,0,0,0,0,0,17,71,0,0,71,56] >;

C3×C83Q8 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_3Q_8
% in TeX

G:=Group("C3xC8:3Q8");
// GroupNames label

G:=SmallGroup(192,931);
// by ID

G=gap.SmallGroup(192,931);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,365,176,1094,268,6053,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations

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