metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊2C8, C12.25Q16, C12.40SD16, C42.193D6, C12.4M4(2), C4⋊C8.5S3, C4.2(S3×C8), C3⋊2(Q8⋊C8), C6.11C4≀C2, C12.4(C2×C8), C2.8(D6⋊C8), C4.2(C8⋊S3), (C2×C4).110D12, (C2×C12).226D4, C6.6(C22⋊C8), C4⋊Dic3.15C4, (C4×Dic6).6C2, C4.15(D4.S3), (C4×C12).42C22, C6.2(Q8⋊C4), (C2×Dic6).11C4, C4.13(C3⋊Q16), C2.2(D12⋊C4), C22.35(D6⋊C4), C2.1(C6.SD16), (C4×C3⋊C8).3C2, (C3×C4⋊C8).5C2, (C2×C4).66(C4×S3), (C2×C12).49(C2×C4), (C2×C4).266(C3⋊D4), (C2×C6).46(C22⋊C4), SmallGroup(192,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊2C8
G = < a,b,c | a12=c8=1, b2=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b >
Subgroups: 168 in 70 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C4×C8, C4⋊C8, C4×Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, Q8⋊C8, C4×C3⋊C8, C3×C4⋊C8, C4×Dic6, Dic6⋊2C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), SD16, Q16, C4×S3, D12, C3⋊D4, C22⋊C8, Q8⋊C4, C4≀C2, S3×C8, C8⋊S3, D6⋊C4, D4.S3, C3⋊Q16, Q8⋊C8, C6.SD16, D6⋊C8, D12⋊C4, Dic6⋊2C8
►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 66 7 72)(2 65 8 71)(3 64 9 70)(4 63 10 69)(5 62 11 68)(6 61 12 67)(13 30 19 36)(14 29 20 35)(15 28 21 34)(16 27 22 33)(17 26 23 32)(18 25 24 31)(37 51 43 57)(38 50 44 56)(39 49 45 55)(40 60 46 54)(41 59 47 53)(42 58 48 52)(73 182 79 188)(74 181 80 187)(75 192 81 186)(76 191 82 185)(77 190 83 184)(78 189 84 183)(85 97 91 103)(86 108 92 102)(87 107 93 101)(88 106 94 100)(89 105 95 99)(90 104 96 98)(109 122 115 128)(110 121 116 127)(111 132 117 126)(112 131 118 125)(113 130 119 124)(114 129 120 123)(133 156 139 150)(134 155 140 149)(135 154 141 148)(136 153 142 147)(137 152 143 146)(138 151 144 145)(157 177 163 171)(158 176 164 170)(159 175 165 169)(160 174 166 180)(161 173 167 179)(162 172 168 178)
(1 179 141 73 92 29 119 43)(2 174 142 80 93 36 120 38)(3 169 143 75 94 31 109 45)(4 176 144 82 95 26 110 40)(5 171 133 77 96 33 111 47)(6 178 134 84 85 28 112 42)(7 173 135 79 86 35 113 37)(8 180 136 74 87 30 114 44)(9 175 137 81 88 25 115 39)(10 170 138 76 89 32 116 46)(11 177 139 83 90 27 117 41)(12 172 140 78 91 34 118 48)(13 126 50 68 166 150 187 104)(14 121 51 63 167 145 188 99)(15 128 52 70 168 152 189 106)(16 123 53 65 157 147 190 101)(17 130 54 72 158 154 191 108)(18 125 55 67 159 149 192 103)(19 132 56 62 160 156 181 98)(20 127 57 69 161 151 182 105)(21 122 58 64 162 146 183 100)(22 129 59 71 163 153 184 107)(23 124 60 66 164 148 185 102)(24 131 49 61 165 155 186 97)
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), (1,66,7,72)(2,65,8,71)(3,64,9,70)(4,63,10,69)(5,62,11,68)(6,61,12,67)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31)(37,51,43,57)(38,50,44,56)(39,49,45,55)(40,60,46,54)(41,59,47,53)(42,58,48,52)(73,182,79,188)(74,181,80,187)(75,192,81,186)(76,191,82,185)(77,190,83,184)(78,189,84,183)(85,97,91,103)(86,108,92,102)(87,107,93,101)(88,106,94,100)(89,105,95,99)(90,104,96,98)(109,122,115,128)(110,121,116,127)(111,132,117,126)(112,131,118,125)(113,130,119,124)(114,129,120,123)(133,156,139,150)(134,155,140,149)(135,154,141,148)(136,153,142,147)(137,152,143,146)(138,151,144,145)(157,177,163,171)(158,176,164,170)(159,175,165,169)(160,174,166,180)(161,173,167,179)(162,172,168,178), (1,179,141,73,92,29,119,43)(2,174,142,80,93,36,120,38)(3,169,143,75,94,31,109,45)(4,176,144,82,95,26,110,40)(5,171,133,77,96,33,111,47)(6,178,134,84,85,28,112,42)(7,173,135,79,86,35,113,37)(8,180,136,74,87,30,114,44)(9,175,137,81,88,25,115,39)(10,170,138,76,89,32,116,46)(11,177,139,83,90,27,117,41)(12,172,140,78,91,34,118,48)(13,126,50,68,166,150,187,104)(14,121,51,63,167,145,188,99)(15,128,52,70,168,152,189,106)(16,123,53,65,157,147,190,101)(17,130,54,72,158,154,191,108)(18,125,55,67,159,149,192,103)(19,132,56,62,160,156,181,98)(20,127,57,69,161,151,182,105)(21,122,58,64,162,146,183,100)(22,129,59,71,163,153,184,107)(23,124,60,66,164,148,185,102)(24,131,49,61,165,155,186,97)>;
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), (1,66,7,72)(2,65,8,71)(3,64,9,70)(4,63,10,69)(5,62,11,68)(6,61,12,67)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31)(37,51,43,57)(38,50,44,56)(39,49,45,55)(40,60,46,54)(41,59,47,53)(42,58,48,52)(73,182,79,188)(74,181,80,187)(75,192,81,186)(76,191,82,185)(77,190,83,184)(78,189,84,183)(85,97,91,103)(86,108,92,102)(87,107,93,101)(88,106,94,100)(89,105,95,99)(90,104,96,98)(109,122,115,128)(110,121,116,127)(111,132,117,126)(112,131,118,125)(113,130,119,124)(114,129,120,123)(133,156,139,150)(134,155,140,149)(135,154,141,148)(136,153,142,147)(137,152,143,146)(138,151,144,145)(157,177,163,171)(158,176,164,170)(159,175,165,169)(160,174,166,180)(161,173,167,179)(162,172,168,178), (1,179,141,73,92,29,119,43)(2,174,142,80,93,36,120,38)(3,169,143,75,94,31,109,45)(4,176,144,82,95,26,110,40)(5,171,133,77,96,33,111,47)(6,178,134,84,85,28,112,42)(7,173,135,79,86,35,113,37)(8,180,136,74,87,30,114,44)(9,175,137,81,88,25,115,39)(10,170,138,76,89,32,116,46)(11,177,139,83,90,27,117,41)(12,172,140,78,91,34,118,48)(13,126,50,68,166,150,187,104)(14,121,51,63,167,145,188,99)(15,128,52,70,168,152,189,106)(16,123,53,65,157,147,190,101)(17,130,54,72,158,154,191,108)(18,125,55,67,159,149,192,103)(19,132,56,62,160,156,181,98)(20,127,57,69,161,151,182,105)(21,122,58,64,162,146,183,100)(22,129,59,71,163,153,184,107)(23,124,60,66,164,148,185,102)(24,131,49,61,165,155,186,97) );
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)], [(1,66,7,72),(2,65,8,71),(3,64,9,70),(4,63,10,69),(5,62,11,68),(6,61,12,67),(13,30,19,36),(14,29,20,35),(15,28,21,34),(16,27,22,33),(17,26,23,32),(18,25,24,31),(37,51,43,57),(38,50,44,56),(39,49,45,55),(40,60,46,54),(41,59,47,53),(42,58,48,52),(73,182,79,188),(74,181,80,187),(75,192,81,186),(76,191,82,185),(77,190,83,184),(78,189,84,183),(85,97,91,103),(86,108,92,102),(87,107,93,101),(88,106,94,100),(89,105,95,99),(90,104,96,98),(109,122,115,128),(110,121,116,127),(111,132,117,126),(112,131,118,125),(113,130,119,124),(114,129,120,123),(133,156,139,150),(134,155,140,149),(135,154,141,148),(136,153,142,147),(137,152,143,146),(138,151,144,145),(157,177,163,171),(158,176,164,170),(159,175,165,169),(160,174,166,180),(161,173,167,179),(162,172,168,178)], [(1,179,141,73,92,29,119,43),(2,174,142,80,93,36,120,38),(3,169,143,75,94,31,109,45),(4,176,144,82,95,26,110,40),(5,171,133,77,96,33,111,47),(6,178,134,84,85,28,112,42),(7,173,135,79,86,35,113,37),(8,180,136,74,87,30,114,44),(9,175,137,81,88,25,115,39),(10,170,138,76,89,32,116,46),(11,177,139,83,90,27,117,41),(12,172,140,78,91,34,118,48),(13,126,50,68,166,150,187,104),(14,121,51,63,167,145,188,99),(15,128,52,70,168,152,189,106),(16,123,53,65,157,147,190,101),(17,130,54,72,158,154,191,108),(18,125,55,67,159,149,192,103),(19,132,56,62,160,156,181,98),(20,127,57,69,161,151,182,105),(21,122,58,64,162,146,183,100),(22,129,59,71,163,153,184,107),(23,124,60,66,164,148,185,102),(24,131,49,61,165,155,186,97)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | S3×C8 | C8⋊S3 | D4.S3 | C3⋊Q16 | D12⋊C4 |
| kernel | Dic6⋊2C8 | C4×C3⋊C8 | C3×C4⋊C8 | C4×Dic6 | C4⋊Dic3 | C2×Dic6 | Dic6 | C4⋊C8 | C2×C12 | C42 | C12 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C6 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of Dic6⋊2C8 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 48 |
| 0 | 0 | 3 | 72 |
| 18 | 23 | 0 | 0 |
| 5 | 55 | 0 | 0 |
| 0 | 0 | 27 | 55 |
| 0 | 0 | 0 | 46 |
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 38 |
| 0 | 0 | 25 | 0 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,1,3,0,0,48,72],[18,5,0,0,23,55,0,0,0,0,27,0,0,0,55,46],[22,0,0,0,0,22,0,0,0,0,0,25,0,0,38,0] >;
Dic6⋊2C8 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_2C_8 % in TeX
G:=Group("Dic6:2C8"); // GroupNames label
G:=SmallGroup(192,43);
// by ID
G=gap.SmallGroup(192,43);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,100,1123,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=c^8=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations