direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C2×C6×C4⋊C4, C4⋊2(C22×C12), C12⋊10(C22×C4), (C22×C4)⋊11C12, (C22×C12)⋊20C4, C2.2(C23×C12), C24.42(C2×C6), C6.54(C23×C4), (C23×C4).11C6, C22.58(C6×D4), C23.64(C3×D4), C6.55(C22×Q8), (C22×C6).30Q8, C23.12(C3×Q8), C22.16(C6×Q8), (C23×C12).10C2, C23.44(C2×C12), (C2×C6).333C24, (C22×C6).220D4, C6.178(C22×D4), C22.6(C23×C6), (C2×C12).958C23, C23.70(C22×C6), (C22×C6).466C23, C22.24(C22×C12), (C23×C6).119C22, (C22×C12).507C22, C2.2(D4×C2×C6), C2.1(Q8×C2×C6), (C2×C4)⋊10(C2×C12), (C2×C12)⋊39(C2×C4), (C2×C6).680(C2×D4), (C2×C6).114(C2×Q8), (C2×C4).53(C22×C6), (C2×C6).245(C22×C4), (C22×C4).130(C2×C6), (C22×C6).148(C2×C4), SmallGroup(192,1402)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6×C4⋊C4
G = < a,b,c,d | a2=b6=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 498 in 418 conjugacy classes, 338 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C23×C4, C23×C4, C3×C4⋊C4, C22×C12, C22×C12, C23×C6, C22×C4⋊C4, C6×C4⋊C4, C23×C12, C23×C12, C2×C6×C4⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C23×C6, C22×C4⋊C4, C6×C4⋊C4, C23×C12, D4×C2×C6, Q8×C2×C6, C2×C6×C4⋊C4
►Regular action on 192 points
Generators in S192
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 166)(8 167)(9 168)(10 163)(11 164)(12 165)(13 177)(14 178)(15 179)(16 180)(17 175)(18 176)(19 171)(20 172)(21 173)(22 174)(23 169)(24 170)(25 58)(26 59)(27 60)(28 55)(29 56)(30 57)(31 66)(32 61)(33 62)(34 63)(35 64)(36 65)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(67 113)(68 114)(69 109)(70 110)(71 111)(72 112)(73 106)(74 107)(75 108)(76 103)(77 104)(78 105)(79 100)(80 101)(81 102)(82 97)(83 98)(84 99)(85 93)(86 94)(87 95)(88 96)(89 91)(90 92)(115 161)(116 162)(117 157)(118 158)(119 159)(120 160)(121 154)(122 155)(123 156)(124 151)(125 152)(126 153)(127 148)(128 149)(129 150)(130 145)(131 146)(132 147)(133 141)(134 142)(135 143)(136 144)(137 139)(138 140)(181 189)(182 190)(183 191)(184 192)(185 187)(186 188)
(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)
(1 101 31 111)(2 102 32 112)(3 97 33 113)(4 98 34 114)(5 99 35 109)(6 100 36 110)(7 115 17 130)(8 116 18 131)(9 117 13 132)(10 118 14 127)(11 119 15 128)(12 120 16 129)(19 126 189 136)(20 121 190 137)(21 122 191 138)(22 123 192 133)(23 124 187 134)(24 125 188 135)(25 103 41 94)(26 104 42 95)(27 105 37 96)(28 106 38 91)(29 107 39 92)(30 108 40 93)(43 86 58 76)(44 87 59 77)(45 88 60 78)(46 89 55 73)(47 90 56 74)(48 85 57 75)(49 79 65 70)(50 80 66 71)(51 81 61 72)(52 82 62 67)(53 83 63 68)(54 84 64 69)(139 172 154 182)(140 173 155 183)(141 174 156 184)(142 169 151 185)(143 170 152 186)(144 171 153 181)(145 166 161 175)(146 167 162 176)(147 168 157 177)(148 163 158 178)(149 164 159 179)(150 165 160 180)
(1 159 39 155)(2 160 40 156)(3 161 41 151)(4 162 42 152)(5 157 37 153)(6 158 38 154)(7 86 23 82)(8 87 24 83)(9 88 19 84)(10 89 20 79)(11 90 21 80)(12 85 22 81)(13 78 189 69)(14 73 190 70)(15 74 191 71)(16 75 192 72)(17 76 187 67)(18 77 188 68)(25 142 33 145)(26 143 34 146)(27 144 35 147)(28 139 36 148)(29 140 31 149)(30 141 32 150)(43 124 52 115)(44 125 53 116)(45 126 54 117)(46 121 49 118)(47 122 50 119)(48 123 51 120)(55 137 65 127)(56 138 66 128)(57 133 61 129)(58 134 62 130)(59 135 63 131)(60 136 64 132)(91 172 100 163)(92 173 101 164)(93 174 102 165)(94 169 97 166)(95 170 98 167)(96 171 99 168)(103 185 113 175)(104 186 114 176)(105 181 109 177)(106 182 110 178)(107 183 111 179)(108 184 112 180)
G:=sub<Sym(192)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,166)(8,167)(9,168)(10,163)(11,164)(12,165)(13,177)(14,178)(15,179)(16,180)(17,175)(18,176)(19,171)(20,172)(21,173)(22,174)(23,169)(24,170)(25,58)(26,59)(27,60)(28,55)(29,56)(30,57)(31,66)(32,61)(33,62)(34,63)(35,64)(36,65)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(67,113)(68,114)(69,109)(70,110)(71,111)(72,112)(73,106)(74,107)(75,108)(76,103)(77,104)(78,105)(79,100)(80,101)(81,102)(82,97)(83,98)(84,99)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92)(115,161)(116,162)(117,157)(118,158)(119,159)(120,160)(121,154)(122,155)(123,156)(124,151)(125,152)(126,153)(127,148)(128,149)(129,150)(130,145)(131,146)(132,147)(133,141)(134,142)(135,143)(136,144)(137,139)(138,140)(181,189)(182,190)(183,191)(184,192)(185,187)(186,188), (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), (1,101,31,111)(2,102,32,112)(3,97,33,113)(4,98,34,114)(5,99,35,109)(6,100,36,110)(7,115,17,130)(8,116,18,131)(9,117,13,132)(10,118,14,127)(11,119,15,128)(12,120,16,129)(19,126,189,136)(20,121,190,137)(21,122,191,138)(22,123,192,133)(23,124,187,134)(24,125,188,135)(25,103,41,94)(26,104,42,95)(27,105,37,96)(28,106,38,91)(29,107,39,92)(30,108,40,93)(43,86,58,76)(44,87,59,77)(45,88,60,78)(46,89,55,73)(47,90,56,74)(48,85,57,75)(49,79,65,70)(50,80,66,71)(51,81,61,72)(52,82,62,67)(53,83,63,68)(54,84,64,69)(139,172,154,182)(140,173,155,183)(141,174,156,184)(142,169,151,185)(143,170,152,186)(144,171,153,181)(145,166,161,175)(146,167,162,176)(147,168,157,177)(148,163,158,178)(149,164,159,179)(150,165,160,180), (1,159,39,155)(2,160,40,156)(3,161,41,151)(4,162,42,152)(5,157,37,153)(6,158,38,154)(7,86,23,82)(8,87,24,83)(9,88,19,84)(10,89,20,79)(11,90,21,80)(12,85,22,81)(13,78,189,69)(14,73,190,70)(15,74,191,71)(16,75,192,72)(17,76,187,67)(18,77,188,68)(25,142,33,145)(26,143,34,146)(27,144,35,147)(28,139,36,148)(29,140,31,149)(30,141,32,150)(43,124,52,115)(44,125,53,116)(45,126,54,117)(46,121,49,118)(47,122,50,119)(48,123,51,120)(55,137,65,127)(56,138,66,128)(57,133,61,129)(58,134,62,130)(59,135,63,131)(60,136,64,132)(91,172,100,163)(92,173,101,164)(93,174,102,165)(94,169,97,166)(95,170,98,167)(96,171,99,168)(103,185,113,175)(104,186,114,176)(105,181,109,177)(106,182,110,178)(107,183,111,179)(108,184,112,180)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,166)(8,167)(9,168)(10,163)(11,164)(12,165)(13,177)(14,178)(15,179)(16,180)(17,175)(18,176)(19,171)(20,172)(21,173)(22,174)(23,169)(24,170)(25,58)(26,59)(27,60)(28,55)(29,56)(30,57)(31,66)(32,61)(33,62)(34,63)(35,64)(36,65)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(67,113)(68,114)(69,109)(70,110)(71,111)(72,112)(73,106)(74,107)(75,108)(76,103)(77,104)(78,105)(79,100)(80,101)(81,102)(82,97)(83,98)(84,99)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92)(115,161)(116,162)(117,157)(118,158)(119,159)(120,160)(121,154)(122,155)(123,156)(124,151)(125,152)(126,153)(127,148)(128,149)(129,150)(130,145)(131,146)(132,147)(133,141)(134,142)(135,143)(136,144)(137,139)(138,140)(181,189)(182,190)(183,191)(184,192)(185,187)(186,188), (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), (1,101,31,111)(2,102,32,112)(3,97,33,113)(4,98,34,114)(5,99,35,109)(6,100,36,110)(7,115,17,130)(8,116,18,131)(9,117,13,132)(10,118,14,127)(11,119,15,128)(12,120,16,129)(19,126,189,136)(20,121,190,137)(21,122,191,138)(22,123,192,133)(23,124,187,134)(24,125,188,135)(25,103,41,94)(26,104,42,95)(27,105,37,96)(28,106,38,91)(29,107,39,92)(30,108,40,93)(43,86,58,76)(44,87,59,77)(45,88,60,78)(46,89,55,73)(47,90,56,74)(48,85,57,75)(49,79,65,70)(50,80,66,71)(51,81,61,72)(52,82,62,67)(53,83,63,68)(54,84,64,69)(139,172,154,182)(140,173,155,183)(141,174,156,184)(142,169,151,185)(143,170,152,186)(144,171,153,181)(145,166,161,175)(146,167,162,176)(147,168,157,177)(148,163,158,178)(149,164,159,179)(150,165,160,180), (1,159,39,155)(2,160,40,156)(3,161,41,151)(4,162,42,152)(5,157,37,153)(6,158,38,154)(7,86,23,82)(8,87,24,83)(9,88,19,84)(10,89,20,79)(11,90,21,80)(12,85,22,81)(13,78,189,69)(14,73,190,70)(15,74,191,71)(16,75,192,72)(17,76,187,67)(18,77,188,68)(25,142,33,145)(26,143,34,146)(27,144,35,147)(28,139,36,148)(29,140,31,149)(30,141,32,150)(43,124,52,115)(44,125,53,116)(45,126,54,117)(46,121,49,118)(47,122,50,119)(48,123,51,120)(55,137,65,127)(56,138,66,128)(57,133,61,129)(58,134,62,130)(59,135,63,131)(60,136,64,132)(91,172,100,163)(92,173,101,164)(93,174,102,165)(94,169,97,166)(95,170,98,167)(96,171,99,168)(103,185,113,175)(104,186,114,176)(105,181,109,177)(106,182,110,178)(107,183,111,179)(108,184,112,180) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,49),(7,166),(8,167),(9,168),(10,163),(11,164),(12,165),(13,177),(14,178),(15,179),(16,180),(17,175),(18,176),(19,171),(20,172),(21,173),(22,174),(23,169),(24,170),(25,58),(26,59),(27,60),(28,55),(29,56),(30,57),(31,66),(32,61),(33,62),(34,63),(35,64),(36,65),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44),(67,113),(68,114),(69,109),(70,110),(71,111),(72,112),(73,106),(74,107),(75,108),(76,103),(77,104),(78,105),(79,100),(80,101),(81,102),(82,97),(83,98),(84,99),(85,93),(86,94),(87,95),(88,96),(89,91),(90,92),(115,161),(116,162),(117,157),(118,158),(119,159),(120,160),(121,154),(122,155),(123,156),(124,151),(125,152),(126,153),(127,148),(128,149),(129,150),(130,145),(131,146),(132,147),(133,141),(134,142),(135,143),(136,144),(137,139),(138,140),(181,189),(182,190),(183,191),(184,192),(185,187),(186,188)], [(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)], [(1,101,31,111),(2,102,32,112),(3,97,33,113),(4,98,34,114),(5,99,35,109),(6,100,36,110),(7,115,17,130),(8,116,18,131),(9,117,13,132),(10,118,14,127),(11,119,15,128),(12,120,16,129),(19,126,189,136),(20,121,190,137),(21,122,191,138),(22,123,192,133),(23,124,187,134),(24,125,188,135),(25,103,41,94),(26,104,42,95),(27,105,37,96),(28,106,38,91),(29,107,39,92),(30,108,40,93),(43,86,58,76),(44,87,59,77),(45,88,60,78),(46,89,55,73),(47,90,56,74),(48,85,57,75),(49,79,65,70),(50,80,66,71),(51,81,61,72),(52,82,62,67),(53,83,63,68),(54,84,64,69),(139,172,154,182),(140,173,155,183),(141,174,156,184),(142,169,151,185),(143,170,152,186),(144,171,153,181),(145,166,161,175),(146,167,162,176),(147,168,157,177),(148,163,158,178),(149,164,159,179),(150,165,160,180)], [(1,159,39,155),(2,160,40,156),(3,161,41,151),(4,162,42,152),(5,157,37,153),(6,158,38,154),(7,86,23,82),(8,87,24,83),(9,88,19,84),(10,89,20,79),(11,90,21,80),(12,85,22,81),(13,78,189,69),(14,73,190,70),(15,74,191,71),(16,75,192,72),(17,76,187,67),(18,77,188,68),(25,142,33,145),(26,143,34,146),(27,144,35,147),(28,139,36,148),(29,140,31,149),(30,141,32,150),(43,124,52,115),(44,125,53,116),(45,126,54,117),(46,121,49,118),(47,122,50,119),(48,123,51,120),(55,137,65,127),(56,138,66,128),(57,133,61,129),(58,134,62,130),(59,135,63,131),(60,136,64,132),(91,172,100,163),(92,173,101,164),(93,174,102,165),(94,169,97,166),(95,170,98,167),(96,171,99,168),(103,185,113,175),(104,186,114,176),(105,181,109,177),(106,182,110,178),(107,183,111,179),(108,184,112,180)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3A | 3B | 4A | ··· | 4X | 6A | ··· | 6AD | 12A | ··· | 12AV |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 |
| kernel | C2×C6×C4⋊C4 | C6×C4⋊C4 | C23×C12 | C22×C4⋊C4 | C22×C12 | C2×C4⋊C4 | C23×C4 | C22×C4 | C22×C6 | C22×C6 | C23 | C23 |
| # reps | 1 | 12 | 3 | 2 | 16 | 24 | 6 | 32 | 4 | 4 | 8 | 8 |
Matrix representation of C2×C6×C4⋊C4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 2 | 7 |
| 0 | 0 | 0 | 7 | 11 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,12,0],[5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,2,7,0,0,0,7,11] >;
C2×C6×C4⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_4\rtimes C_4
% in TeX
G:=Group("C2xC6xC4:C4"); // GroupNames label
G:=SmallGroup(192,1402);
// by ID
G=gap.SmallGroup(192,1402);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,344]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations