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## G = C42×C12order 192 = 26·3

### Abelian group of type [4,4,12]

Aliases: C42×C12, SmallGroup(192,807)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42×C12
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C2×C4×C12 — C42×C12
 Lower central C1 — C42×C12
 Upper central C1 — C42×C12

Generators and relations for C42×C12
G = < a,b,c | a4=b4=c12=1, ab=ba, ac=ca, bc=cb >

Subgroups: 258, all normal (6 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, C2×C12, C22×C6, C2×C42, C4×C12, C22×C12, C43, C2×C4×C12, C42×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, C2×C12, C22×C6, C2×C42, C4×C12, C22×C12, C43, C2×C4×C12, C42×C12

Smallest permutation representation of C42×C12
Regular action on 192 points
Generators in S192
(1 85 108 186)(2 86 97 187)(3 87 98 188)(4 88 99 189)(5 89 100 190)(6 90 101 191)(7 91 102 192)(8 92 103 181)(9 93 104 182)(10 94 105 183)(11 95 106 184)(12 96 107 185)(13 149 143 75)(14 150 144 76)(15 151 133 77)(16 152 134 78)(17 153 135 79)(18 154 136 80)(19 155 137 81)(20 156 138 82)(21 145 139 83)(22 146 140 84)(23 147 141 73)(24 148 142 74)(25 46 116 166)(26 47 117 167)(27 48 118 168)(28 37 119 157)(29 38 120 158)(30 39 109 159)(31 40 110 160)(32 41 111 161)(33 42 112 162)(34 43 113 163)(35 44 114 164)(36 45 115 165)(49 176 129 71)(50 177 130 72)(51 178 131 61)(52 179 132 62)(53 180 121 63)(54 169 122 64)(55 170 123 65)(56 171 124 66)(57 172 125 67)(58 173 126 68)(59 174 127 69)(60 175 128 70)
(1 157 61 80)(2 158 62 81)(3 159 63 82)(4 160 64 83)(5 161 65 84)(6 162 66 73)(7 163 67 74)(8 164 68 75)(9 165 69 76)(10 166 70 77)(11 167 71 78)(12 168 72 79)(13 92 35 58)(14 93 36 59)(15 94 25 60)(16 95 26 49)(17 96 27 50)(18 85 28 51)(19 86 29 52)(20 87 30 53)(21 88 31 54)(22 89 32 55)(23 90 33 56)(24 91 34 57)(37 178 154 108)(38 179 155 97)(39 180 156 98)(40 169 145 99)(41 170 146 100)(42 171 147 101)(43 172 148 102)(44 173 149 103)(45 174 150 104)(46 175 151 105)(47 176 152 106)(48 177 153 107)(109 121 138 188)(110 122 139 189)(111 123 140 190)(112 124 141 191)(113 125 142 192)(114 126 143 181)(115 127 144 182)(116 128 133 183)(117 129 134 184)(118 130 135 185)(119 131 136 186)(120 132 137 187)
(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)| (1,85,108,186)(2,86,97,187)(3,87,98,188)(4,88,99,189)(5,89,100,190)(6,90,101,191)(7,91,102,192)(8,92,103,181)(9,93,104,182)(10,94,105,183)(11,95,106,184)(12,96,107,185)(13,149,143,75)(14,150,144,76)(15,151,133,77)(16,152,134,78)(17,153,135,79)(18,154,136,80)(19,155,137,81)(20,156,138,82)(21,145,139,83)(22,146,140,84)(23,147,141,73)(24,148,142,74)(25,46,116,166)(26,47,117,167)(27,48,118,168)(28,37,119,157)(29,38,120,158)(30,39,109,159)(31,40,110,160)(32,41,111,161)(33,42,112,162)(34,43,113,163)(35,44,114,164)(36,45,115,165)(49,176,129,71)(50,177,130,72)(51,178,131,61)(52,179,132,62)(53,180,121,63)(54,169,122,64)(55,170,123,65)(56,171,124,66)(57,172,125,67)(58,173,126,68)(59,174,127,69)(60,175,128,70), (1,157,61,80)(2,158,62,81)(3,159,63,82)(4,160,64,83)(5,161,65,84)(6,162,66,73)(7,163,67,74)(8,164,68,75)(9,165,69,76)(10,166,70,77)(11,167,71,78)(12,168,72,79)(13,92,35,58)(14,93,36,59)(15,94,25,60)(16,95,26,49)(17,96,27,50)(18,85,28,51)(19,86,29,52)(20,87,30,53)(21,88,31,54)(22,89,32,55)(23,90,33,56)(24,91,34,57)(37,178,154,108)(38,179,155,97)(39,180,156,98)(40,169,145,99)(41,170,146,100)(42,171,147,101)(43,172,148,102)(44,173,149,103)(45,174,150,104)(46,175,151,105)(47,176,152,106)(48,177,153,107)(109,121,138,188)(110,122,139,189)(111,123,140,190)(112,124,141,191)(113,125,142,192)(114,126,143,181)(115,127,144,182)(116,128,133,183)(117,129,134,184)(118,130,135,185)(119,131,136,186)(120,132,137,187), (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( (1,85,108,186)(2,86,97,187)(3,87,98,188)(4,88,99,189)(5,89,100,190)(6,90,101,191)(7,91,102,192)(8,92,103,181)(9,93,104,182)(10,94,105,183)(11,95,106,184)(12,96,107,185)(13,149,143,75)(14,150,144,76)(15,151,133,77)(16,152,134,78)(17,153,135,79)(18,154,136,80)(19,155,137,81)(20,156,138,82)(21,145,139,83)(22,146,140,84)(23,147,141,73)(24,148,142,74)(25,46,116,166)(26,47,117,167)(27,48,118,168)(28,37,119,157)(29,38,120,158)(30,39,109,159)(31,40,110,160)(32,41,111,161)(33,42,112,162)(34,43,113,163)(35,44,114,164)(36,45,115,165)(49,176,129,71)(50,177,130,72)(51,178,131,61)(52,179,132,62)(53,180,121,63)(54,169,122,64)(55,170,123,65)(56,171,124,66)(57,172,125,67)(58,173,126,68)(59,174,127,69)(60,175,128,70), (1,157,61,80)(2,158,62,81)(3,159,63,82)(4,160,64,83)(5,161,65,84)(6,162,66,73)(7,163,67,74)(8,164,68,75)(9,165,69,76)(10,166,70,77)(11,167,71,78)(12,168,72,79)(13,92,35,58)(14,93,36,59)(15,94,25,60)(16,95,26,49)(17,96,27,50)(18,85,28,51)(19,86,29,52)(20,87,30,53)(21,88,31,54)(22,89,32,55)(23,90,33,56)(24,91,34,57)(37,178,154,108)(38,179,155,97)(39,180,156,98)(40,169,145,99)(41,170,146,100)(42,171,147,101)(43,172,148,102)(44,173,149,103)(45,174,150,104)(46,175,151,105)(47,176,152,106)(48,177,153,107)(109,121,138,188)(110,122,139,189)(111,123,140,190)(112,124,141,191)(113,125,142,192)(114,126,143,181)(115,127,144,182)(116,128,133,183)(117,129,134,184)(118,130,135,185)(119,131,136,186)(120,132,137,187), (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([[(1,85,108,186),(2,86,97,187),(3,87,98,188),(4,88,99,189),(5,89,100,190),(6,90,101,191),(7,91,102,192),(8,92,103,181),(9,93,104,182),(10,94,105,183),(11,95,106,184),(12,96,107,185),(13,149,143,75),(14,150,144,76),(15,151,133,77),(16,152,134,78),(17,153,135,79),(18,154,136,80),(19,155,137,81),(20,156,138,82),(21,145,139,83),(22,146,140,84),(23,147,141,73),(24,148,142,74),(25,46,116,166),(26,47,117,167),(27,48,118,168),(28,37,119,157),(29,38,120,158),(30,39,109,159),(31,40,110,160),(32,41,111,161),(33,42,112,162),(34,43,113,163),(35,44,114,164),(36,45,115,165),(49,176,129,71),(50,177,130,72),(51,178,131,61),(52,179,132,62),(53,180,121,63),(54,169,122,64),(55,170,123,65),(56,171,124,66),(57,172,125,67),(58,173,126,68),(59,174,127,69),(60,175,128,70)], [(1,157,61,80),(2,158,62,81),(3,159,63,82),(4,160,64,83),(5,161,65,84),(6,162,66,73),(7,163,67,74),(8,164,68,75),(9,165,69,76),(10,166,70,77),(11,167,71,78),(12,168,72,79),(13,92,35,58),(14,93,36,59),(15,94,25,60),(16,95,26,49),(17,96,27,50),(18,85,28,51),(19,86,29,52),(20,87,30,53),(21,88,31,54),(22,89,32,55),(23,90,33,56),(24,91,34,57),(37,178,154,108),(38,179,155,97),(39,180,156,98),(40,169,145,99),(41,170,146,100),(42,171,147,101),(43,172,148,102),(44,173,149,103),(45,174,150,104),(46,175,151,105),(47,176,152,106),(48,177,153,107),(109,121,138,188),(110,122,139,189),(111,123,140,190),(112,124,141,191),(113,125,142,192),(114,126,143,181),(115,127,144,182),(116,128,133,183),(117,129,134,184),(118,130,135,185),(119,131,136,186),(120,132,137,187)], [(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)]])

192 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4BD 6A ··· 6N 12A ··· 12DH order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

192 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C12 kernel C42×C12 C2×C4×C12 C43 C4×C12 C2×C42 C42 # reps 1 7 2 56 14 112

Matrix representation of C42×C12 in GL3(𝔽13) generated by

 8 0 0 0 8 0 0 0 8
,
 5 0 0 0 12 0 0 0 5
,
 5 0 0 0 10 0 0 0 10
G:=sub<GL(3,GF(13))| [8,0,0,0,8,0,0,0,8],[5,0,0,0,12,0,0,0,5],[5,0,0,0,10,0,0,0,10] >;

C42×C12 in GAP, Magma, Sage, TeX

C_4^2\times C_{12}
% in TeX

G:=Group("C4^2xC12");
// GroupNames label

G:=SmallGroup(192,807);
// by ID

G=gap.SmallGroup(192,807);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,344,520]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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