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G = C42×C12order 192 = 26·3

Abelian group of type [4,4,12]

direct product, abelian, monomial, 2-elementary

Aliases: C42×C12, SmallGroup(192,807)

Series: Derived Chief Lower central Upper central

C1 — C42×C12
C1C2C22C23C22×C6C22×C12C2×C4×C12 — C42×C12
C1 — C42×C12
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 [×7], C3, C4 [×28], C22 [×7], C6 [×7], C2×C4 [×42], C23, C12 [×28], C2×C6 [×7], C42 [×28], C22×C4 [×7], C2×C12 [×42], C22×C6, C2×C42 [×7], C4×C12 [×28], C22×C12 [×7], C43, C2×C4×C12 [×7], C42×C12
Quotients: C1, C2 [×7], C3, C4 [×28], C22 [×7], C6 [×7], C2×C4 [×42], C23, C12 [×28], C2×C6 [×7], C42 [×28], C22×C4 [×7], C2×C12 [×42], C22×C6, C2×C42 [×7], C4×C12 [×28], C22×C12 [×7], C43, C2×C4×C12 [×7], C42×C12

Smallest permutation representation of C42×C12
Regular action on 192 points
Generators in S192
(1 150 74 55)(2 151 75 56)(3 152 76 57)(4 153 77 58)(5 154 78 59)(6 155 79 60)(7 156 80 49)(8 145 81 50)(9 146 82 51)(10 147 83 52)(11 148 84 53)(12 149 73 54)(13 28 37 165)(14 29 38 166)(15 30 39 167)(16 31 40 168)(17 32 41 157)(18 33 42 158)(19 34 43 159)(20 35 44 160)(21 36 45 161)(22 25 46 162)(23 26 47 163)(24 27 48 164)(61 192 121 115)(62 181 122 116)(63 182 123 117)(64 183 124 118)(65 184 125 119)(66 185 126 120)(67 186 127 109)(68 187 128 110)(69 188 129 111)(70 189 130 112)(71 190 131 113)(72 191 132 114)(85 99 141 172)(86 100 142 173)(87 101 143 174)(88 102 144 175)(89 103 133 176)(90 104 134 177)(91 105 135 178)(92 106 136 179)(93 107 137 180)(94 108 138 169)(95 97 139 170)(96 98 140 171)
(1 167 65 179)(2 168 66 180)(3 157 67 169)(4 158 68 170)(5 159 69 171)(6 160 70 172)(7 161 71 173)(8 162 72 174)(9 163 61 175)(10 164 62 176)(11 165 63 177)(12 166 64 178)(13 182 90 148)(14 183 91 149)(15 184 92 150)(16 185 93 151)(17 186 94 152)(18 187 95 153)(19 188 96 154)(20 189 85 155)(21 190 86 156)(22 191 87 145)(23 192 88 146)(24 181 89 147)(25 132 101 81)(26 121 102 82)(27 122 103 83)(28 123 104 84)(29 124 105 73)(30 125 106 74)(31 126 107 75)(32 127 108 76)(33 128 97 77)(34 129 98 78)(35 130 99 79)(36 131 100 80)(37 117 134 53)(38 118 135 54)(39 119 136 55)(40 120 137 56)(41 109 138 57)(42 110 139 58)(43 111 140 59)(44 112 141 60)(45 113 142 49)(46 114 143 50)(47 115 144 51)(48 116 133 52)
(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)| (1,150,74,55)(2,151,75,56)(3,152,76,57)(4,153,77,58)(5,154,78,59)(6,155,79,60)(7,156,80,49)(8,145,81,50)(9,146,82,51)(10,147,83,52)(11,148,84,53)(12,149,73,54)(13,28,37,165)(14,29,38,166)(15,30,39,167)(16,31,40,168)(17,32,41,157)(18,33,42,158)(19,34,43,159)(20,35,44,160)(21,36,45,161)(22,25,46,162)(23,26,47,163)(24,27,48,164)(61,192,121,115)(62,181,122,116)(63,182,123,117)(64,183,124,118)(65,184,125,119)(66,185,126,120)(67,186,127,109)(68,187,128,110)(69,188,129,111)(70,189,130,112)(71,190,131,113)(72,191,132,114)(85,99,141,172)(86,100,142,173)(87,101,143,174)(88,102,144,175)(89,103,133,176)(90,104,134,177)(91,105,135,178)(92,106,136,179)(93,107,137,180)(94,108,138,169)(95,97,139,170)(96,98,140,171), (1,167,65,179)(2,168,66,180)(3,157,67,169)(4,158,68,170)(5,159,69,171)(6,160,70,172)(7,161,71,173)(8,162,72,174)(9,163,61,175)(10,164,62,176)(11,165,63,177)(12,166,64,178)(13,182,90,148)(14,183,91,149)(15,184,92,150)(16,185,93,151)(17,186,94,152)(18,187,95,153)(19,188,96,154)(20,189,85,155)(21,190,86,156)(22,191,87,145)(23,192,88,146)(24,181,89,147)(25,132,101,81)(26,121,102,82)(27,122,103,83)(28,123,104,84)(29,124,105,73)(30,125,106,74)(31,126,107,75)(32,127,108,76)(33,128,97,77)(34,129,98,78)(35,130,99,79)(36,131,100,80)(37,117,134,53)(38,118,135,54)(39,119,136,55)(40,120,137,56)(41,109,138,57)(42,110,139,58)(43,111,140,59)(44,112,141,60)(45,113,142,49)(46,114,143,50)(47,115,144,51)(48,116,133,52), (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( (1,150,74,55)(2,151,75,56)(3,152,76,57)(4,153,77,58)(5,154,78,59)(6,155,79,60)(7,156,80,49)(8,145,81,50)(9,146,82,51)(10,147,83,52)(11,148,84,53)(12,149,73,54)(13,28,37,165)(14,29,38,166)(15,30,39,167)(16,31,40,168)(17,32,41,157)(18,33,42,158)(19,34,43,159)(20,35,44,160)(21,36,45,161)(22,25,46,162)(23,26,47,163)(24,27,48,164)(61,192,121,115)(62,181,122,116)(63,182,123,117)(64,183,124,118)(65,184,125,119)(66,185,126,120)(67,186,127,109)(68,187,128,110)(69,188,129,111)(70,189,130,112)(71,190,131,113)(72,191,132,114)(85,99,141,172)(86,100,142,173)(87,101,143,174)(88,102,144,175)(89,103,133,176)(90,104,134,177)(91,105,135,178)(92,106,136,179)(93,107,137,180)(94,108,138,169)(95,97,139,170)(96,98,140,171), (1,167,65,179)(2,168,66,180)(3,157,67,169)(4,158,68,170)(5,159,69,171)(6,160,70,172)(7,161,71,173)(8,162,72,174)(9,163,61,175)(10,164,62,176)(11,165,63,177)(12,166,64,178)(13,182,90,148)(14,183,91,149)(15,184,92,150)(16,185,93,151)(17,186,94,152)(18,187,95,153)(19,188,96,154)(20,189,85,155)(21,190,86,156)(22,191,87,145)(23,192,88,146)(24,181,89,147)(25,132,101,81)(26,121,102,82)(27,122,103,83)(28,123,104,84)(29,124,105,73)(30,125,106,74)(31,126,107,75)(32,127,108,76)(33,128,97,77)(34,129,98,78)(35,130,99,79)(36,131,100,80)(37,117,134,53)(38,118,135,54)(39,119,136,55)(40,120,137,56)(41,109,138,57)(42,110,139,58)(43,111,140,59)(44,112,141,60)(45,113,142,49)(46,114,143,50)(47,115,144,51)(48,116,133,52), (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([(1,150,74,55),(2,151,75,56),(3,152,76,57),(4,153,77,58),(5,154,78,59),(6,155,79,60),(7,156,80,49),(8,145,81,50),(9,146,82,51),(10,147,83,52),(11,148,84,53),(12,149,73,54),(13,28,37,165),(14,29,38,166),(15,30,39,167),(16,31,40,168),(17,32,41,157),(18,33,42,158),(19,34,43,159),(20,35,44,160),(21,36,45,161),(22,25,46,162),(23,26,47,163),(24,27,48,164),(61,192,121,115),(62,181,122,116),(63,182,123,117),(64,183,124,118),(65,184,125,119),(66,185,126,120),(67,186,127,109),(68,187,128,110),(69,188,129,111),(70,189,130,112),(71,190,131,113),(72,191,132,114),(85,99,141,172),(86,100,142,173),(87,101,143,174),(88,102,144,175),(89,103,133,176),(90,104,134,177),(91,105,135,178),(92,106,136,179),(93,107,137,180),(94,108,138,169),(95,97,139,170),(96,98,140,171)], [(1,167,65,179),(2,168,66,180),(3,157,67,169),(4,158,68,170),(5,159,69,171),(6,160,70,172),(7,161,71,173),(8,162,72,174),(9,163,61,175),(10,164,62,176),(11,165,63,177),(12,166,64,178),(13,182,90,148),(14,183,91,149),(15,184,92,150),(16,185,93,151),(17,186,94,152),(18,187,95,153),(19,188,96,154),(20,189,85,155),(21,190,86,156),(22,191,87,145),(23,192,88,146),(24,181,89,147),(25,132,101,81),(26,121,102,82),(27,122,103,83),(28,123,104,84),(29,124,105,73),(30,125,106,74),(31,126,107,75),(32,127,108,76),(33,128,97,77),(34,129,98,78),(35,130,99,79),(36,131,100,80),(37,117,134,53),(38,118,135,54),(39,119,136,55),(40,120,137,56),(41,109,138,57),(42,110,139,58),(43,111,140,59),(44,112,141,60),(45,113,142,49),(46,114,143,50),(47,115,144,51),(48,116,133,52)], [(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···2G3A3B4A···4BD6A···6N12A···12DH
order12···2334···46···612···12
size11···1111···11···11···1

192 irreducible representations

dim111111
type++
imageC1C2C3C4C6C12
kernelC42×C12C2×C4×C12C43C4×C12C2×C42C42
# reps1725614112

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

800
080
008
,
500
0120
005
,
500
0100
0010
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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