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## G = C4×Dic3⋊C4order 192 = 26·3

### Direct product of C4 and Dic3⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×Dic3⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C4×Dic3⋊C4
 Lower central C3 — C6 — C4×Dic3⋊C4
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C4×Dic3⋊C4
G = < a,b,c,d | a4=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 376 in 194 conjugacy classes, 107 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×14], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×22], C23, Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×4], C2×C6 [×3], C2×C6 [×4], C42 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×12], C2×Dic3 [×6], C2×C12 [×10], C2×C12 [×4], C22×C6, C2.C42 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C4×Dic3 [×4], C4×Dic3 [×2], Dic3⋊C4 [×8], C4×C12 [×2], C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C4×C4⋊C4, C6.C42 [×2], C2×C4×Dic3 [×2], C2×Dic3⋊C4 [×2], C2×C4×C12, C4×Dic3⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], D4 [×2], Q8 [×2], C23, D6 [×3], C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], Dic6 [×2], C4×S3 [×6], C3⋊D4 [×2], C22×S3, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4 [×3], C4○D12 [×2], C2×C3⋊D4, C4×C4⋊C4, C4×Dic6 [×2], S3×C42, C422S3, C2×Dic3⋊C4, C4×C3⋊D4 [×2], C4×Dic3⋊C4

Smallest permutation representation of C4×Dic3⋊C4
Regular action on 192 points
Generators in S192
(1 58 7 52)(2 59 8 53)(3 60 9 54)(4 55 10 49)(5 56 11 50)(6 57 12 51)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 67 22 61)(17 68 23 62)(18 69 24 63)(25 82 31 76)(26 83 32 77)(27 84 33 78)(28 79 34 73)(29 80 35 74)(30 81 36 75)(37 94 43 88)(38 95 44 89)(39 96 45 90)(40 91 46 85)(41 92 47 86)(42 93 48 87)(97 154 103 148)(98 155 104 149)(99 156 105 150)(100 151 106 145)(101 152 107 146)(102 153 108 147)(109 166 115 160)(110 167 116 161)(111 168 117 162)(112 163 118 157)(113 164 119 158)(114 165 120 159)(121 178 127 172)(122 179 128 173)(123 180 129 174)(124 175 130 169)(125 176 131 170)(126 177 132 171)(133 190 139 184)(134 191 140 185)(135 192 141 186)(136 187 142 181)(137 188 143 182)(138 189 144 183)
(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)
(1 110 4 113)(2 109 5 112)(3 114 6 111)(7 116 10 119)(8 115 11 118)(9 120 12 117)(13 98 16 101)(14 97 17 100)(15 102 18 99)(19 104 22 107)(20 103 23 106)(21 108 24 105)(25 134 28 137)(26 133 29 136)(27 138 30 135)(31 140 34 143)(32 139 35 142)(33 144 36 141)(37 122 40 125)(38 121 41 124)(39 126 42 123)(43 128 46 131)(44 127 47 130)(45 132 48 129)(49 158 52 161)(50 157 53 160)(51 162 54 159)(55 164 58 167)(56 163 59 166)(57 168 60 165)(61 146 64 149)(62 145 65 148)(63 150 66 147)(67 152 70 155)(68 151 71 154)(69 156 72 153)(73 182 76 185)(74 181 77 184)(75 186 78 183)(79 188 82 191)(80 187 83 190)(81 192 84 189)(85 170 88 173)(86 169 89 172)(87 174 90 171)(91 176 94 179)(92 175 95 178)(93 180 96 177)
(1 91 19 73)(2 92 20 74)(3 93 21 75)(4 94 22 76)(5 95 23 77)(6 96 24 78)(7 85 13 79)(8 86 14 80)(9 87 15 81)(10 88 16 82)(11 89 17 83)(12 90 18 84)(25 55 43 61)(26 56 44 62)(27 57 45 63)(28 58 46 64)(29 59 47 65)(30 60 48 66)(31 49 37 67)(32 50 38 68)(33 51 39 69)(34 52 40 70)(35 53 41 71)(36 54 42 72)(97 190 115 172)(98 191 116 173)(99 192 117 174)(100 187 118 169)(101 188 119 170)(102 189 120 171)(103 184 109 178)(104 185 110 179)(105 186 111 180)(106 181 112 175)(107 182 113 176)(108 183 114 177)(121 154 139 160)(122 155 140 161)(123 156 141 162)(124 151 142 157)(125 152 143 158)(126 153 144 159)(127 148 133 166)(128 149 134 167)(129 150 135 168)(130 145 136 163)(131 146 137 164)(132 147 138 165)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + - image C1 C2 C2 C2 C2 C4 C4 S3 D4 Q8 D6 C4○D4 Dic6 C4×S3 C3⋊D4 C4○D12 kernel C4×Dic3⋊C4 C6.C42 C2×C4×Dic3 C2×Dic3⋊C4 C2×C4×C12 C4×Dic3 Dic3⋊C4 C2×C42 C2×C12 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 2 2 1 8 16 1 2 2 3 4 4 12 4 8

Matrix representation of C4×Dic3⋊C4 in GL4(𝔽13) generated by

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

C4×Dic3⋊C4 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("C4xDic3:C4");
// GroupNames label

G:=SmallGroup(192,490);
// by ID

G=gap.SmallGroup(192,490);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,758,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^6=d^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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