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## G = C12×C4⋊C4order 192 = 26·3

### Direct product of C12 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C12×C4⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C6 — C22×C12 — C3×C2.C42 — C12×C4⋊C4
 Lower central C1 — C2 — C12×C4⋊C4
 Upper central C1 — C22×C12 — C12×C4⋊C4

Generators and relations for C12×C4⋊C4
G = < a,b,c | a12=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 242 in 194 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C22×C12, C22×C12, C4×C4⋊C4, C3×C2.C42, C2×C4×C12, C2×C4×C12, C6×C4⋊C4, C12×C4⋊C4
Quotients:

Smallest permutation representation of C12×C4⋊C4
Regular action on 192 points
Generators in S192
(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)
(1 176 83 137)(2 177 84 138)(3 178 73 139)(4 179 74 140)(5 180 75 141)(6 169 76 142)(7 170 77 143)(8 171 78 144)(9 172 79 133)(10 173 80 134)(11 174 81 135)(12 175 82 136)(13 107 37 117)(14 108 38 118)(15 97 39 119)(16 98 40 120)(17 99 41 109)(18 100 42 110)(19 101 43 111)(20 102 44 112)(21 103 45 113)(22 104 46 114)(23 105 47 115)(24 106 48 116)(25 125 191 86)(26 126 192 87)(27 127 181 88)(28 128 182 89)(29 129 183 90)(30 130 184 91)(31 131 185 92)(32 132 186 93)(33 121 187 94)(34 122 188 95)(35 123 189 96)(36 124 190 85)(49 164 71 154)(50 165 72 155)(51 166 61 156)(52 167 62 145)(53 168 63 146)(54 157 64 147)(55 158 65 148)(56 159 66 149)(57 160 67 150)(58 161 68 151)(59 162 69 152)(60 163 70 153)
(1 89 69 114)(2 90 70 115)(3 91 71 116)(4 92 72 117)(5 93 61 118)(6 94 62 119)(7 95 63 120)(8 96 64 109)(9 85 65 110)(10 86 66 111)(11 87 67 112)(12 88 68 113)(13 140 31 165)(14 141 32 166)(15 142 33 167)(16 143 34 168)(17 144 35 157)(18 133 36 158)(19 134 25 159)(20 135 26 160)(21 136 27 161)(22 137 28 162)(23 138 29 163)(24 139 30 164)(37 179 185 155)(38 180 186 156)(39 169 187 145)(40 170 188 146)(41 171 189 147)(42 172 190 148)(43 173 191 149)(44 174 192 150)(45 175 181 151)(46 176 182 152)(47 177 183 153)(48 178 184 154)(49 106 73 130)(50 107 74 131)(51 108 75 132)(52 97 76 121)(53 98 77 122)(54 99 78 123)(55 100 79 124)(56 101 80 125)(57 102 81 126)(58 103 82 127)(59 104 83 128)(60 105 84 129)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 4I ··· 4AF 6A ··· 6N 12A ··· 12P 12Q ··· 12BL order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 Q8 C4○D4 C3×D4 C3×Q8 C3×C4○D4 kernel C12×C4⋊C4 C3×C2.C42 C2×C4×C12 C6×C4⋊C4 C4×C4⋊C4 C4×C12 C3×C4⋊C4 C2.C42 C2×C42 C2×C4⋊C4 C42 C4⋊C4 C2×C12 C2×C12 C2×C6 C2×C4 C2×C4 C22 # reps 1 2 3 2 2 8 16 4 6 4 16 32 2 2 4 4 4 8

Matrix representation of C12×C4⋊C4 in GL4(𝔽13) generated by

 12 0 0 0 0 8 0 0 0 0 2 0 0 0 0 2
,
 12 0 0 0 0 12 0 0 0 0 7 9 0 0 6 6
,
 8 0 0 0 0 1 0 0 0 0 5 10 0 0 0 8
G:=sub<GL(4,GF(13))| [12,0,0,0,0,8,0,0,0,0,2,0,0,0,0,2],[12,0,0,0,0,12,0,0,0,0,7,6,0,0,9,6],[8,0,0,0,0,1,0,0,0,0,5,0,0,0,10,8] >;

C12×C4⋊C4 in GAP, Magma, Sage, TeX

C_{12}\times C_4\rtimes C_4
% in TeX

G:=Group("C12xC4:C4");
// GroupNames label

G:=SmallGroup(192,811);
// by ID

G=gap.SmallGroup(192,811);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,394]);
// Polycyclic

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

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