direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3×C4⋊C4, C12⋊C42, C4⋊1(C4×Dic3), C6.58(C4×D4), C6.21(C4×Q8), C4⋊Dic3⋊13C4, (C4×Dic3)⋊7C4, C2.1(Q8×Dic3), C2.2(D4×Dic3), C6.20(C2×C42), C22.20(S3×Q8), (C2×Dic3).22Q8, C22.105(S3×D4), (C22×C4).349D6, C2.3(Dic3⋊5D4), (C2×Dic3).211D4, C6.34(C42⋊C2), C2.3(Dic6⋊C4), C6.C42.34C2, C23.298(C22×S3), (C22×C6).338C23, C22.53(D4⋊2S3), (C22×C12).345C22, C22.22(Q8⋊3S3), C22.20(C22×Dic3), (C22×Dic3).187C22, C3⋊4(C4×C4⋊C4), (C3×C4⋊C4)⋊7C4, C2.4(S3×C4⋊C4), C6.18(C2×C4⋊C4), (C6×C4⋊C4).11C2, (C2×C4⋊C4).28S3, C2.9(C2×C4×Dic3), C22.55(S3×C2×C4), (C2×C6).74(C2×Q8), (C2×C4×Dic3).5C2, (C2×C4).151(C4×S3), (C2×C12).82(C2×C4), (C2×C6).328(C2×D4), C2.4(C4⋊C4⋊7S3), (C2×C4⋊Dic3).32C2, (C2×C4).32(C2×Dic3), (C2×C6).150(C4○D4), (C2×C6).114(C22×C4), (C2×Dic3).61(C2×C4), SmallGroup(192,533)
Series: Derived ►Chief ►Lower central ►Upper central
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, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 376 in 194 conjugacy classes, 111 normal (41 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Dic3, C4×Dic3, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C4×C4⋊C4, C6.C42, C2×C4×Dic3, C2×C4×Dic3, C2×C4⋊Dic3, C6×C4⋊C4, Dic3×C4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C2×Dic3, C22×S3, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Dic3, S3×C2×C4, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×Dic3, C4×C4⋊C4, Dic6⋊C4, S3×C4⋊C4, C4⋊C4⋊7S3, Dic3⋊5D4, C2×C4×Dic3, D4×Dic3, Q8×Dic3, 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)
(1 104 4 107)(2 103 5 106)(3 108 6 105)(7 114 10 111)(8 113 11 110)(9 112 12 109)(13 102 16 99)(14 101 17 98)(15 100 18 97)(19 116 22 119)(20 115 23 118)(21 120 24 117)(25 125 28 122)(26 124 29 121)(27 123 30 126)(31 131 34 128)(32 130 35 127)(33 129 36 132)(37 134 40 137)(38 133 41 136)(39 138 42 135)(43 140 46 143)(44 139 47 142)(45 144 48 141)(49 149 52 146)(50 148 53 145)(51 147 54 150)(55 155 58 152)(56 154 59 151)(57 153 60 156)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 164 70 167)(68 163 71 166)(69 168 72 165)(73 173 76 170)(74 172 77 169)(75 171 78 174)(79 179 82 176)(80 178 83 175)(81 177 84 180)(85 182 88 185)(86 181 89 184)(87 186 90 183)(91 188 94 191)(92 187 95 190)(93 192 96 189)
(1 76 23 62)(2 77 24 63)(3 78 19 64)(4 73 20 65)(5 74 21 66)(6 75 22 61)(7 72 29 59)(8 67 30 60)(9 68 25 55)(10 69 26 56)(11 70 27 57)(12 71 28 58)(13 142 181 128)(14 143 182 129)(15 144 183 130)(16 139 184 131)(17 140 185 132)(18 141 186 127)(31 102 44 89)(32 97 45 90)(33 98 46 85)(34 99 47 86)(35 100 48 87)(36 101 43 88)(37 94 51 81)(38 95 52 82)(39 96 53 83)(40 91 54 84)(41 92 49 79)(42 93 50 80)(103 169 117 162)(104 170 118 157)(105 171 119 158)(106 172 120 159)(107 173 115 160)(108 174 116 161)(109 166 122 152)(110 167 123 153)(111 168 124 154)(112 163 125 155)(113 164 126 156)(114 165 121 151)(133 190 146 176)(134 191 147 177)(135 192 148 178)(136 187 149 179)(137 188 150 180)(138 189 145 175)
(1 38 12 34)(2 39 7 35)(3 40 8 36)(4 41 9 31)(5 42 10 32)(6 37 11 33)(13 157 190 152)(14 158 191 153)(15 159 192 154)(16 160 187 155)(17 161 188 156)(18 162 189 151)(19 54 30 43)(20 49 25 44)(21 50 26 45)(22 51 27 46)(23 52 28 47)(24 53 29 48)(55 102 65 92)(56 97 66 93)(57 98 61 94)(58 99 62 95)(59 100 63 96)(60 101 64 91)(67 88 78 84)(68 89 73 79)(69 90 74 80)(70 85 75 81)(71 86 76 82)(72 87 77 83)(103 138 114 127)(104 133 109 128)(105 134 110 129)(106 135 111 130)(107 136 112 131)(108 137 113 132)(115 149 125 139)(116 150 126 140)(117 145 121 141)(118 146 122 142)(119 147 123 143)(120 148 124 144)(163 184 173 179)(164 185 174 180)(165 186 169 175)(166 181 170 176)(167 182 171 177)(168 183 172 178)
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), (1,104,4,107)(2,103,5,106)(3,108,6,105)(7,114,10,111)(8,113,11,110)(9,112,12,109)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189), (1,76,23,62)(2,77,24,63)(3,78,19,64)(4,73,20,65)(5,74,21,66)(6,75,22,61)(7,72,29,59)(8,67,30,60)(9,68,25,55)(10,69,26,56)(11,70,27,57)(12,71,28,58)(13,142,181,128)(14,143,182,129)(15,144,183,130)(16,139,184,131)(17,140,185,132)(18,141,186,127)(31,102,44,89)(32,97,45,90)(33,98,46,85)(34,99,47,86)(35,100,48,87)(36,101,43,88)(37,94,51,81)(38,95,52,82)(39,96,53,83)(40,91,54,84)(41,92,49,79)(42,93,50,80)(103,169,117,162)(104,170,118,157)(105,171,119,158)(106,172,120,159)(107,173,115,160)(108,174,116,161)(109,166,122,152)(110,167,123,153)(111,168,124,154)(112,163,125,155)(113,164,126,156)(114,165,121,151)(133,190,146,176)(134,191,147,177)(135,192,148,178)(136,187,149,179)(137,188,150,180)(138,189,145,175), (1,38,12,34)(2,39,7,35)(3,40,8,36)(4,41,9,31)(5,42,10,32)(6,37,11,33)(13,157,190,152)(14,158,191,153)(15,159,192,154)(16,160,187,155)(17,161,188,156)(18,162,189,151)(19,54,30,43)(20,49,25,44)(21,50,26,45)(22,51,27,46)(23,52,28,47)(24,53,29,48)(55,102,65,92)(56,97,66,93)(57,98,61,94)(58,99,62,95)(59,100,63,96)(60,101,64,91)(67,88,78,84)(68,89,73,79)(69,90,74,80)(70,85,75,81)(71,86,76,82)(72,87,77,83)(103,138,114,127)(104,133,109,128)(105,134,110,129)(106,135,111,130)(107,136,112,131)(108,137,113,132)(115,149,125,139)(116,150,126,140)(117,145,121,141)(118,146,122,142)(119,147,123,143)(120,148,124,144)(163,184,173,179)(164,185,174,180)(165,186,169,175)(166,181,170,176)(167,182,171,177)(168,183,172,178)>;
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), (1,104,4,107)(2,103,5,106)(3,108,6,105)(7,114,10,111)(8,113,11,110)(9,112,12,109)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189), (1,76,23,62)(2,77,24,63)(3,78,19,64)(4,73,20,65)(5,74,21,66)(6,75,22,61)(7,72,29,59)(8,67,30,60)(9,68,25,55)(10,69,26,56)(11,70,27,57)(12,71,28,58)(13,142,181,128)(14,143,182,129)(15,144,183,130)(16,139,184,131)(17,140,185,132)(18,141,186,127)(31,102,44,89)(32,97,45,90)(33,98,46,85)(34,99,47,86)(35,100,48,87)(36,101,43,88)(37,94,51,81)(38,95,52,82)(39,96,53,83)(40,91,54,84)(41,92,49,79)(42,93,50,80)(103,169,117,162)(104,170,118,157)(105,171,119,158)(106,172,120,159)(107,173,115,160)(108,174,116,161)(109,166,122,152)(110,167,123,153)(111,168,124,154)(112,163,125,155)(113,164,126,156)(114,165,121,151)(133,190,146,176)(134,191,147,177)(135,192,148,178)(136,187,149,179)(137,188,150,180)(138,189,145,175), (1,38,12,34)(2,39,7,35)(3,40,8,36)(4,41,9,31)(5,42,10,32)(6,37,11,33)(13,157,190,152)(14,158,191,153)(15,159,192,154)(16,160,187,155)(17,161,188,156)(18,162,189,151)(19,54,30,43)(20,49,25,44)(21,50,26,45)(22,51,27,46)(23,52,28,47)(24,53,29,48)(55,102,65,92)(56,97,66,93)(57,98,61,94)(58,99,62,95)(59,100,63,96)(60,101,64,91)(67,88,78,84)(68,89,73,79)(69,90,74,80)(70,85,75,81)(71,86,76,82)(72,87,77,83)(103,138,114,127)(104,133,109,128)(105,134,110,129)(106,135,111,130)(107,136,112,131)(108,137,113,132)(115,149,125,139)(116,150,126,140)(117,145,121,141)(118,146,122,142)(119,147,123,143)(120,148,124,144)(163,184,173,179)(164,185,174,180)(165,186,169,175)(166,181,170,176)(167,182,171,177)(168,183,172,178) );
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)], [(1,104,4,107),(2,103,5,106),(3,108,6,105),(7,114,10,111),(8,113,11,110),(9,112,12,109),(13,102,16,99),(14,101,17,98),(15,100,18,97),(19,116,22,119),(20,115,23,118),(21,120,24,117),(25,125,28,122),(26,124,29,121),(27,123,30,126),(31,131,34,128),(32,130,35,127),(33,129,36,132),(37,134,40,137),(38,133,41,136),(39,138,42,135),(43,140,46,143),(44,139,47,142),(45,144,48,141),(49,149,52,146),(50,148,53,145),(51,147,54,150),(55,155,58,152),(56,154,59,151),(57,153,60,156),(61,158,64,161),(62,157,65,160),(63,162,66,159),(67,164,70,167),(68,163,71,166),(69,168,72,165),(73,173,76,170),(74,172,77,169),(75,171,78,174),(79,179,82,176),(80,178,83,175),(81,177,84,180),(85,182,88,185),(86,181,89,184),(87,186,90,183),(91,188,94,191),(92,187,95,190),(93,192,96,189)], [(1,76,23,62),(2,77,24,63),(3,78,19,64),(4,73,20,65),(5,74,21,66),(6,75,22,61),(7,72,29,59),(8,67,30,60),(9,68,25,55),(10,69,26,56),(11,70,27,57),(12,71,28,58),(13,142,181,128),(14,143,182,129),(15,144,183,130),(16,139,184,131),(17,140,185,132),(18,141,186,127),(31,102,44,89),(32,97,45,90),(33,98,46,85),(34,99,47,86),(35,100,48,87),(36,101,43,88),(37,94,51,81),(38,95,52,82),(39,96,53,83),(40,91,54,84),(41,92,49,79),(42,93,50,80),(103,169,117,162),(104,170,118,157),(105,171,119,158),(106,172,120,159),(107,173,115,160),(108,174,116,161),(109,166,122,152),(110,167,123,153),(111,168,124,154),(112,163,125,155),(113,164,126,156),(114,165,121,151),(133,190,146,176),(134,191,147,177),(135,192,148,178),(136,187,149,179),(137,188,150,180),(138,189,145,175)], [(1,38,12,34),(2,39,7,35),(3,40,8,36),(4,41,9,31),(5,42,10,32),(6,37,11,33),(13,157,190,152),(14,158,191,153),(15,159,192,154),(16,160,187,155),(17,161,188,156),(18,162,189,151),(19,54,30,43),(20,49,25,44),(21,50,26,45),(22,51,27,46),(23,52,28,47),(24,53,29,48),(55,102,65,92),(56,97,66,93),(57,98,61,94),(58,99,62,95),(59,100,63,96),(60,101,64,91),(67,88,78,84),(68,89,73,79),(69,90,74,80),(70,85,75,81),(71,86,76,82),(72,87,77,83),(103,138,114,127),(104,133,109,128),(105,134,110,129),(106,135,111,130),(107,136,112,131),(108,137,113,132),(115,149,125,139),(116,150,126,140),(117,145,121,141),(118,146,122,142),(119,147,123,143),(120,148,124,144),(163,184,173,179),(164,185,174,180),(165,186,169,175),(166,181,170,176),(167,182,171,177),(168,183,172,178)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AF | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | C4○D4 | C4×S3 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | Dic3×C4⋊C4 | C6.C42 | C2×C4×Dic3 | C2×C4⋊Dic3 | C6×C4⋊C4 | C4×Dic3 | C4⋊Dic3 | C3×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×Dic3 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 3 | 1 | 1 | 8 | 8 | 8 | 1 | 2 | 2 | 4 | 3 | 4 | 8 | 1 | 1 | 1 | 1 |
Matrix representation of Dic3×C4⋊C4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 2 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 4 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,7,0,0,0,0,10,3],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,6,1,0,0,0,0,2,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,4,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
Dic3×C4⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_4\rtimes C_4 % in TeX
G:=Group("Dic3xC4:C4"); // GroupNames label
G:=SmallGroup(192,533);
// by ID
G=gap.SmallGroup(192,533);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,387,100,6278]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations