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G = C3×C4.6Q16order 192 = 26·3

Direct product of C3 and C4.6Q16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C4.6Q16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C12 — C3×C4⋊C8 — C3×C4.6Q16
 Lower central C1 — C22 — C2×C4 — C3×C4.6Q16
 Upper central C1 — C2×C6 — C4×C12 — C3×C4.6Q16

Generators and relations for C3×C4.6Q16
G = < a,b,c,d | a3=b4=c8=1, d2=b2c4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c-1 >

Subgroups: 114 in 64 conjugacy classes, 38 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, C2×C8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C4⋊C8, C4⋊Q8, C4×C12, C3×C4⋊C4, C2×C24, C6×Q8, C4.6Q16, C3×C4⋊C8, C3×C4⋊Q8, C3×C4.6Q16
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, SD16, Q16, C2×C12, C3×D4, C4.D4, Q8⋊C4, C3×C22⋊C4, C3×SD16, C3×Q16, C4.6Q16, C3×C4.D4, C3×Q8⋊C4, C3×C4.6Q16

Smallest permutation representation of C3×C4.6Q16
Regular action on 192 points
Generators in S192
(1 167 159)(2 168 160)(3 161 153)(4 162 154)(5 163 155)(6 164 156)(7 165 157)(8 166 158)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(13 29 21)(14 30 22)(15 31 23)(16 32 24)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(37 53 45)(38 54 46)(39 55 47)(40 56 48)(57 73 65)(58 74 66)(59 75 67)(60 76 68)(61 77 69)(62 78 70)(63 79 71)(64 80 72)(81 97 89)(82 98 90)(83 99 91)(84 100 92)(85 101 93)(86 102 94)(87 103 95)(88 104 96)(105 121 113)(106 122 114)(107 123 115)(108 124 116)(109 125 117)(110 126 118)(111 127 119)(112 128 120)(129 145 137)(130 146 138)(131 147 139)(132 148 140)(133 149 141)(134 150 142)(135 151 143)(136 152 144)(169 185 177)(170 186 178)(171 187 179)(172 188 180)(173 189 181)(174 190 182)(175 191 183)(176 192 184)
(1 57 11 109)(2 110 12 58)(3 59 13 111)(4 112 14 60)(5 61 15 105)(6 106 16 62)(7 63 9 107)(8 108 10 64)(17 115 157 71)(18 72 158 116)(19 117 159 65)(20 66 160 118)(21 119 153 67)(22 68 154 120)(23 113 155 69)(24 70 156 114)(25 123 165 79)(26 80 166 124)(27 125 167 73)(28 74 168 126)(29 127 161 75)(30 76 162 128)(31 121 163 77)(32 78 164 122)(33 83 173 135)(34 136 174 84)(35 85 175 129)(36 130 176 86)(37 87 169 131)(38 132 170 88)(39 81 171 133)(40 134 172 82)(41 91 181 143)(42 144 182 92)(43 93 183 137)(44 138 184 94)(45 95 177 139)(46 140 178 96)(47 89 179 141)(48 142 180 90)(49 99 189 151)(50 152 190 100)(51 101 191 145)(52 146 192 102)(53 103 185 147)(54 148 186 104)(55 97 187 149)(56 150 188 98)
(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 35 15 171)(2 136 16 88)(3 33 9 169)(4 134 10 86)(5 39 11 175)(6 132 12 84)(7 37 13 173)(8 130 14 82)(17 177 153 41)(18 94 154 142)(19 183 155 47)(20 92 156 140)(21 181 157 45)(22 90 158 138)(23 179 159 43)(24 96 160 144)(25 185 161 49)(26 102 162 150)(27 191 163 55)(28 100 164 148)(29 189 165 53)(30 98 166 146)(31 187 167 51)(32 104 168 152)(34 62 170 110)(36 60 172 108)(38 58 174 106)(40 64 176 112)(42 70 178 118)(44 68 180 116)(46 66 182 114)(48 72 184 120)(50 78 186 126)(52 76 188 124)(54 74 190 122)(56 80 192 128)(57 129 105 81)(59 135 107 87)(61 133 109 85)(63 131 111 83)(65 137 113 89)(67 143 115 95)(69 141 117 93)(71 139 119 91)(73 145 121 97)(75 151 123 103)(77 149 125 101)(79 147 127 99)

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 D4 SD16 Q16 C3×D4 C3×SD16 C3×Q16 C4.D4 C3×C4.D4 kernel C3×C4.6Q16 C3×C4⋊C8 C3×C4⋊Q8 C4.6Q16 C6×Q8 C4⋊C8 C4⋊Q8 C2×Q8 C2×C12 C12 C12 C2×C4 C4 C4 C6 C2 # reps 1 2 1 2 4 4 2 8 2 4 4 4 8 8 1 2

Matrix representation of C3×C4.6Q16 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 64 0 0 0 0 64
,
 0 72 0 0 1 0 0 0 0 0 72 0 0 0 0 72
,
 52 61 0 0 61 21 0 0 0 0 0 6 0 0 61 12
,
 19 21 0 0 21 54 0 0 0 0 60 12 0 0 71 13
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[0,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[52,61,0,0,61,21,0,0,0,0,0,61,0,0,6,12],[19,21,0,0,21,54,0,0,0,0,60,71,0,0,12,13] >;

C3×C4.6Q16 in GAP, Magma, Sage, TeX

C_3\times C_4._6Q_{16}
% in TeX

G:=Group("C3xC4.6Q16");
// GroupNames label

G:=SmallGroup(192,139);
// by ID

G=gap.SmallGroup(192,139);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,1522,248,2951,242]);
// Polycyclic

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

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