metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊1C8, C12.16Q16, C4.8Dic12, C12.27SD16, C42.246D6, C12.12M4(2), C6.3C4≀C2, C4.6(S3×C8), (C4×C8).2S3, C3⋊1(Q8⋊C8), (C4×C24).2C2, C12.16(C2×C8), C2.3(D6⋊C8), C4⋊Dic3.4C4, C12⋊C8.1C2, C4.4(C8⋊S3), (C2×C12).435D4, (C2×C4).158D12, C6.1(C22⋊C8), (C2×Dic6).5C4, (C4×Dic6).1C2, C4.14(C24⋊C2), C6.5(Q8⋊C4), C2.1(C42⋊4S3), (C4×C12).317C22, C22.30(D6⋊C4), C2.1(C2.Dic12), (C2×C4).95(C4×S3), (C2×C12).215(C2×C4), (C2×C4).204(C3⋊D4), (C2×C6).35(C22⋊C4), SmallGroup(192,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.8Dic12
G = < a,b,c | a4=b24=1, c2=a-1b12, ab=ba, ac=ca, cbc-1=ab-1 >
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, C12⋊C8, C4×C24, C4×Dic6, C4.8Dic12
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, C24⋊C2, Dic12, D6⋊C4, Q8⋊C8, C42⋊4S3, C2.Dic12, D6⋊C8, C4.8Dic12
►Regular action on 192 points
Generators in S192
(1 52 112 25)(2 53 113 26)(3 54 114 27)(4 55 115 28)(5 56 116 29)(6 57 117 30)(7 58 118 31)(8 59 119 32)(9 60 120 33)(10 61 97 34)(11 62 98 35)(12 63 99 36)(13 64 100 37)(14 65 101 38)(15 66 102 39)(16 67 103 40)(17 68 104 41)(18 69 105 42)(19 70 106 43)(20 71 107 44)(21 72 108 45)(22 49 109 46)(23 50 110 47)(24 51 111 48)(73 152 121 174)(74 153 122 175)(75 154 123 176)(76 155 124 177)(77 156 125 178)(78 157 126 179)(79 158 127 180)(80 159 128 181)(81 160 129 182)(82 161 130 183)(83 162 131 184)(84 163 132 185)(85 164 133 186)(86 165 134 187)(87 166 135 188)(88 167 136 189)(89 168 137 190)(90 145 138 191)(91 146 139 192)(92 147 140 169)(93 148 141 170)(94 149 142 171)(95 150 143 172)(96 151 144 173)
(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 92 37 181 112 140 64 159)(2 146 38 79 113 192 65 127)(3 138 39 157 114 90 66 179)(4 190 40 125 115 168 67 77)(5 88 41 177 116 136 68 155)(6 166 42 75 117 188 69 123)(7 134 43 153 118 86 70 175)(8 186 44 121 119 164 71 73)(9 84 45 173 120 132 72 151)(10 162 46 95 97 184 49 143)(11 130 47 149 98 82 50 171)(12 182 48 141 99 160 51 93)(13 80 25 169 100 128 52 147)(14 158 26 91 101 180 53 139)(15 126 27 145 102 78 54 191)(16 178 28 137 103 156 55 89)(17 76 29 189 104 124 56 167)(18 154 30 87 105 176 57 135)(19 122 31 165 106 74 58 187)(20 174 32 133 107 152 59 85)(21 96 33 185 108 144 60 163)(22 150 34 83 109 172 61 131)(23 142 35 161 110 94 62 183)(24 170 36 129 111 148 63 81)
G:=sub<Sym(192)| (1,52,112,25)(2,53,113,26)(3,54,114,27)(4,55,115,28)(5,56,116,29)(6,57,117,30)(7,58,118,31)(8,59,119,32)(9,60,120,33)(10,61,97,34)(11,62,98,35)(12,63,99,36)(13,64,100,37)(14,65,101,38)(15,66,102,39)(16,67,103,40)(17,68,104,41)(18,69,105,42)(19,70,106,43)(20,71,107,44)(21,72,108,45)(22,49,109,46)(23,50,110,47)(24,51,111,48)(73,152,121,174)(74,153,122,175)(75,154,123,176)(76,155,124,177)(77,156,125,178)(78,157,126,179)(79,158,127,180)(80,159,128,181)(81,160,129,182)(82,161,130,183)(83,162,131,184)(84,163,132,185)(85,164,133,186)(86,165,134,187)(87,166,135,188)(88,167,136,189)(89,168,137,190)(90,145,138,191)(91,146,139,192)(92,147,140,169)(93,148,141,170)(94,149,142,171)(95,150,143,172)(96,151,144,173), (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), (1,92,37,181,112,140,64,159)(2,146,38,79,113,192,65,127)(3,138,39,157,114,90,66,179)(4,190,40,125,115,168,67,77)(5,88,41,177,116,136,68,155)(6,166,42,75,117,188,69,123)(7,134,43,153,118,86,70,175)(8,186,44,121,119,164,71,73)(9,84,45,173,120,132,72,151)(10,162,46,95,97,184,49,143)(11,130,47,149,98,82,50,171)(12,182,48,141,99,160,51,93)(13,80,25,169,100,128,52,147)(14,158,26,91,101,180,53,139)(15,126,27,145,102,78,54,191)(16,178,28,137,103,156,55,89)(17,76,29,189,104,124,56,167)(18,154,30,87,105,176,57,135)(19,122,31,165,106,74,58,187)(20,174,32,133,107,152,59,85)(21,96,33,185,108,144,60,163)(22,150,34,83,109,172,61,131)(23,142,35,161,110,94,62,183)(24,170,36,129,111,148,63,81)>;
G:=Group( (1,52,112,25)(2,53,113,26)(3,54,114,27)(4,55,115,28)(5,56,116,29)(6,57,117,30)(7,58,118,31)(8,59,119,32)(9,60,120,33)(10,61,97,34)(11,62,98,35)(12,63,99,36)(13,64,100,37)(14,65,101,38)(15,66,102,39)(16,67,103,40)(17,68,104,41)(18,69,105,42)(19,70,106,43)(20,71,107,44)(21,72,108,45)(22,49,109,46)(23,50,110,47)(24,51,111,48)(73,152,121,174)(74,153,122,175)(75,154,123,176)(76,155,124,177)(77,156,125,178)(78,157,126,179)(79,158,127,180)(80,159,128,181)(81,160,129,182)(82,161,130,183)(83,162,131,184)(84,163,132,185)(85,164,133,186)(86,165,134,187)(87,166,135,188)(88,167,136,189)(89,168,137,190)(90,145,138,191)(91,146,139,192)(92,147,140,169)(93,148,141,170)(94,149,142,171)(95,150,143,172)(96,151,144,173), (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), (1,92,37,181,112,140,64,159)(2,146,38,79,113,192,65,127)(3,138,39,157,114,90,66,179)(4,190,40,125,115,168,67,77)(5,88,41,177,116,136,68,155)(6,166,42,75,117,188,69,123)(7,134,43,153,118,86,70,175)(8,186,44,121,119,164,71,73)(9,84,45,173,120,132,72,151)(10,162,46,95,97,184,49,143)(11,130,47,149,98,82,50,171)(12,182,48,141,99,160,51,93)(13,80,25,169,100,128,52,147)(14,158,26,91,101,180,53,139)(15,126,27,145,102,78,54,191)(16,178,28,137,103,156,55,89)(17,76,29,189,104,124,56,167)(18,154,30,87,105,176,57,135)(19,122,31,165,106,74,58,187)(20,174,32,133,107,152,59,85)(21,96,33,185,108,144,60,163)(22,150,34,83,109,172,61,131)(23,142,35,161,110,94,62,183)(24,170,36,129,111,148,63,81) );
G=PermutationGroup([[(1,52,112,25),(2,53,113,26),(3,54,114,27),(4,55,115,28),(5,56,116,29),(6,57,117,30),(7,58,118,31),(8,59,119,32),(9,60,120,33),(10,61,97,34),(11,62,98,35),(12,63,99,36),(13,64,100,37),(14,65,101,38),(15,66,102,39),(16,67,103,40),(17,68,104,41),(18,69,105,42),(19,70,106,43),(20,71,107,44),(21,72,108,45),(22,49,109,46),(23,50,110,47),(24,51,111,48),(73,152,121,174),(74,153,122,175),(75,154,123,176),(76,155,124,177),(77,156,125,178),(78,157,126,179),(79,158,127,180),(80,159,128,181),(81,160,129,182),(82,161,130,183),(83,162,131,184),(84,163,132,185),(85,164,133,186),(86,165,134,187),(87,166,135,188),(88,167,136,189),(89,168,137,190),(90,145,138,191),(91,146,139,192),(92,147,140,169),(93,148,141,170),(94,149,142,171),(95,150,143,172),(96,151,144,173)], [(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)], [(1,92,37,181,112,140,64,159),(2,146,38,79,113,192,65,127),(3,138,39,157,114,90,66,179),(4,190,40,125,115,168,67,77),(5,88,41,177,116,136,68,155),(6,166,42,75,117,188,69,123),(7,134,43,153,118,86,70,175),(8,186,44,121,119,164,71,73),(9,84,45,173,120,132,72,151),(10,162,46,95,97,184,49,143),(11,130,47,149,98,82,50,171),(12,182,48,141,99,160,51,93),(13,80,25,169,100,128,52,147),(14,158,26,91,101,180,53,139),(15,126,27,145,102,78,54,191),(16,178,28,137,103,156,55,89),(17,76,29,189,104,124,56,167),(18,154,30,87,105,176,57,135),(19,122,31,165,106,74,58,187),(20,174,32,133,107,152,59,85),(21,96,33,185,108,144,60,163),(22,150,34,83,109,172,61,131),(23,142,35,161,110,94,62,183),(24,170,36,129,111,148,63,81)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12L | 24A | ··· | 24P |
| 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 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| 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 | C24⋊C2 | Dic12 | C42⋊4S3 |
| kernel | C4.8Dic12 | C12⋊C8 | C4×C24 | 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 | 4 | 4 | 8 |
Matrix representation of C4.8Dic12 ►in GL4(𝔽73) generated by
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 36 | 48 |
| 0 | 0 | 25 | 11 |
| 3 | 68 | 0 | 0 |
| 11 | 70 | 0 | 0 |
| 0 | 0 | 60 | 2 |
| 0 | 0 | 62 | 13 |
G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,72,0,0,0,0,72],[0,46,0,0,1,0,0,0,0,0,36,25,0,0,48,11],[3,11,0,0,68,70,0,0,0,0,60,62,0,0,2,13] >;
C4.8Dic12 in GAP, Magma, Sage, TeX
C_4._8{\rm Dic}_{12} % in TeX
G:=Group("C4.8Dic12"); // GroupNames label
G:=SmallGroup(192,15);
// by ID
G=gap.SmallGroup(192,15);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,85,92,422,100,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=1,c^2=a^-1*b^12,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations