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## G = Dic3⋊(C4⋊C4)  order 192 = 26·3

### 3rd semidirect product of Dic3 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3⋊(C4⋊C4)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — Dic3⋊(C4⋊C4)
 Lower central C3 — C2×C6 — Dic3⋊(C4⋊C4)
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for Dic3⋊(C4⋊C4)
G = < a,b,c,d | a6=c4=d4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 376 in 170 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C3, C4 [×14], C22 [×7], C6 [×7], C2×C4 [×4], C2×C4 [×24], C23, Dic3 [×4], Dic3 [×5], C12 [×5], C2×C6 [×7], C42 [×2], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×10], C2×Dic3 [×7], C2×C12 [×4], C2×C12 [×7], C22×C6, C2.C42 [×2], C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4×Dic3 [×2], Dic3⋊C4 [×4], Dic3⋊C4 [×4], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.65C23, C6.C42 [×2], C2×C4×Dic3, C2×Dic3⋊C4 [×3], C6×C4⋊C4, Dic3⋊(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], Q8 [×4], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, S3×C2×C4, C4○D12, S3×D4, D42S3, S3×Q8 [×2], C2×C3⋊D4, C23.65C23, Dic6⋊C4, Dic3.Q8, S3×C4⋊C4, D6⋊Q8, C4×C3⋊D4, C23.14D6, Dic3⋊Q8, Dic3⋊(C4⋊C4)

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

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

Matrix representation of Dic3⋊(C4⋊C4) in GL7(𝔽13)

 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 10 6 0 0 0 0 0 7 3 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12

G:=sub<GL(7,GF(13))| [1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1],[1,0,0,0,0,0,0,0,10,7,0,0,0,0,0,6,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12] >;

Dic3⋊(C4⋊C4) in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes (C_4\rtimes C_4)
% in TeX

G:=Group("Dic3:(C4:C4)");
// GroupNames label

G:=SmallGroup(192,535);
// by ID

G=gap.SmallGroup(192,535);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,120,219,58,6278]);
// Polycyclic

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

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