Copied to
clipboard

## G = Dic3⋊C4⋊C4order 192 = 26·3

### 3rd semidirect product of Dic3⋊C4 and C4 acting via C4/C2=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×C4×Dic3 — Dic3⋊C4⋊C4
 Lower central C3 — C2×C6 — Dic3⋊C4⋊C4
 Upper central C1 — C23 — C2.C42

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, dbd-1=bc2, dcd-1=a3c-1 >

Subgroups: 352 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×C12, C23.63C23, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, Dic3⋊C4⋊C4
Quotients:

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

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

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

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

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

Matrix representation of Dic3⋊C4⋊C4 in GL6(𝔽13)

 0 1 0 0 0 0 12 1 0 0 0 0 0 0 3 0 0 0 0 0 10 9 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 2 2 0 0 0 0 4 11 0 0 0 0 0 0 11 4 0 0 0 0 9 2 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 3 7 0 0 0 0 6 10 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 2 6 0 0 0 0 6 11
,
 2 9 0 0 0 0 4 11 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,3,10,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[2,4,0,0,0,0,2,11,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,6,0,0,0,0,6,11],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

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(192,214);
// by ID

G=gap.SmallGroup(192,214);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,64,926,219,268,6278]);
// 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,d*b*d^-1=b*c^2,d*c*d^-1=a^3*c^-1>;
// generators/relations

׿
×
𝔽