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## G = C3×C8⋊4Q8order 192 = 26·3

### Direct product of C3 and C8⋊4Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C8⋊4Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C2×C24 — C3×C4⋊C8 — C3×C8⋊4Q8
 Lower central C1 — C22 — C3×C8⋊4Q8
 Upper central C1 — C2×C12 — C3×C8⋊4Q8

Generators and relations for C3×C84Q8
G = < a,b,c,d | a3=b8=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c-1 >

Subgroups: 114 in 94 conjugacy classes, 74 normal (38 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C24, C24, C2×C12, C2×C12, C3×Q8, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C6×Q8, C84Q8, C4×C24, C3×C8⋊C4, C3×C4⋊C8, C3×C4⋊C8, Q8×C12, C3×C84Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, Q8, C23, C12, C2×C6, M4(2), C22×C4, C2×Q8, C4○D4, C2×C12, C3×Q8, C22×C6, C4×Q8, C2×M4(2), C8○D4, C3×M4(2), C22×C12, C6×Q8, C3×C4○D4, C84Q8, Q8×C12, C6×M4(2), C3×C8○D4, C3×C84Q8

Smallest permutation representation of C3×C84Q8
Regular action on 192 points
Generators in S192
(1 55 47)(2 56 48)(3 49 41)(4 50 42)(5 51 43)(6 52 44)(7 53 45)(8 54 46)(9 37 17)(10 38 18)(11 39 19)(12 40 20)(13 33 21)(14 34 22)(15 35 23)(16 36 24)(25 185 168)(26 186 161)(27 187 162)(28 188 163)(29 189 164)(30 190 165)(31 191 166)(32 192 167)(57 78 82)(58 79 83)(59 80 84)(60 73 85)(61 74 86)(62 75 87)(63 76 88)(64 77 81)(65 104 89)(66 97 90)(67 98 91)(68 99 92)(69 100 93)(70 101 94)(71 102 95)(72 103 96)(105 125 130)(106 126 131)(107 127 132)(108 128 133)(109 121 134)(110 122 135)(111 123 136)(112 124 129)(113 145 138)(114 146 139)(115 147 140)(116 148 141)(117 149 142)(118 150 143)(119 151 144)(120 152 137)(153 173 178)(154 174 179)(155 175 180)(156 176 181)(157 169 182)(158 170 183)(159 171 184)(160 172 177)
(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 87 21 71)(2 88 22 72)(3 81 23 65)(4 82 24 66)(5 83 17 67)(6 84 18 68)(7 85 19 69)(8 86 20 70)(9 98 51 58)(10 99 52 59)(11 100 53 60)(12 101 54 61)(13 102 55 62)(14 103 56 63)(15 104 49 64)(16 97 50 57)(25 128 176 152)(26 121 169 145)(27 122 170 146)(28 123 171 147)(29 124 172 148)(30 125 173 149)(31 126 174 150)(32 127 175 151)(33 95 47 75)(34 96 48 76)(35 89 41 77)(36 90 42 78)(37 91 43 79)(38 92 44 80)(39 93 45 73)(40 94 46 74)(105 153 117 165)(106 154 118 166)(107 155 119 167)(108 156 120 168)(109 157 113 161)(110 158 114 162)(111 159 115 163)(112 160 116 164)(129 177 141 189)(130 178 142 190)(131 179 143 191)(132 180 144 192)(133 181 137 185)(134 182 138 186)(135 183 139 187)(136 184 140 188)
(1 105 21 117)(2 110 22 114)(3 107 23 119)(4 112 24 116)(5 109 17 113)(6 106 18 118)(7 111 19 115)(8 108 20 120)(9 145 51 121)(10 150 52 126)(11 147 53 123)(12 152 54 128)(13 149 55 125)(14 146 56 122)(15 151 49 127)(16 148 50 124)(25 101 176 61)(26 98 169 58)(27 103 170 63)(28 100 171 60)(29 97 172 57)(30 102 173 62)(31 99 174 59)(32 104 175 64)(33 142 47 130)(34 139 48 135)(35 144 41 132)(36 141 42 129)(37 138 43 134)(38 143 44 131)(39 140 45 136)(40 137 46 133)(65 155 81 167)(66 160 82 164)(67 157 83 161)(68 154 84 166)(69 159 85 163)(70 156 86 168)(71 153 87 165)(72 158 88 162)(73 188 93 184)(74 185 94 181)(75 190 95 178)(76 187 96 183)(77 192 89 180)(78 189 90 177)(79 186 91 182)(80 191 92 179)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6F 8A ··· 8H 8I 8J 8K 8L 12A ··· 12H 12I ··· 12P 12Q ··· 12X 24A ··· 24P 24Q ··· 24X order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 8 8 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 size 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C2 C3 C4 C4 C6 C6 C6 C6 C12 C12 Q8 M4(2) C4○D4 C3×Q8 C8○D4 C3×M4(2) C3×C4○D4 C3×C8○D4 kernel C3×C8⋊4Q8 C4×C24 C3×C8⋊C4 C3×C4⋊C8 Q8×C12 C8⋊4Q8 C3×C4⋊C4 C6×Q8 C4×C8 C8⋊C4 C4⋊C8 C4×Q8 C4⋊C4 C2×Q8 C24 C12 C12 C8 C6 C4 C4 C2 # reps 1 1 2 3 1 2 6 2 2 4 6 2 12 4 2 4 2 4 4 8 4 8

Matrix representation of C3×C84Q8 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 64 0 0 0 0 64
,
 72 0 0 0 0 72 0 0 0 0 72 71 0 0 14 1
,
 0 1 0 0 72 0 0 0 0 0 72 0 0 0 0 72
,
 54 52 0 0 52 19 0 0 0 0 20 34 0 0 72 53
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,72,14,0,0,71,1],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[54,52,0,0,52,19,0,0,0,0,20,72,0,0,34,53] >;

C3×C84Q8 in GAP, Magma, Sage, TeX

C_3\times C_8\rtimes_4Q_8
% in TeX

G:=Group("C3xC8:4Q8");
// GroupNames label

G:=SmallGroup(192,879);
// by ID

G=gap.SmallGroup(192,879);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,2102,394,124]);
// 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^5,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽