metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.38D4, C22.7D42, C23.2D21, C22⋊2Dic21, (C2×C42)⋊2C4, (C2×C6)⋊2Dic7, C21⋊7(C22⋊C4), C42.32(C2×C4), (C2×C14)⋊4Dic3, (C2×C14).25D6, (C2×C6).25D14, C3⋊2(C23.D7), (C22×C6).2D7, (C2×Dic21)⋊2C2, C2.3(C21⋊7D4), C6.20(C7⋊D4), C7⋊2(C6.D4), (C22×C42).2C2, (C22×C14).4S3, C2.5(C2×Dic21), C6.10(C2×Dic7), C14.20(C3⋊D4), (C2×C42).26C22, C14.10(C2×Dic3), SmallGroup(336,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.38D4
G = < a,b,c | a42=b4=1, c2=a21, bab-1=cac-1=a-1, cbc-1=a21b-1 >
Subgroups: 288 in 68 conjugacy classes, 35 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C7, C2×C4, C23, Dic3, C2×C6, C2×C6, C2×C6, C14, C14, C14, C22⋊C4, C21, C2×Dic3, C22×C6, Dic7, C2×C14, C2×C14, C2×C14, C42, C42, C42, C6.D4, C2×Dic7, C22×C14, Dic21, C2×C42, C2×C42, C2×C42, C23.D7, C2×Dic21, C22×C42, C42.38D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, D7, C22⋊C4, C2×Dic3, C3⋊D4, Dic7, D14, D21, C6.D4, C2×Dic7, C7⋊D4, Dic21, D42, C23.D7, C2×Dic21, C21⋊7D4, C42.38D4
►On 168 points
Generators in S168
(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)
(1 124 154 62)(2 123 155 61)(3 122 156 60)(4 121 157 59)(5 120 158 58)(6 119 159 57)(7 118 160 56)(8 117 161 55)(9 116 162 54)(10 115 163 53)(11 114 164 52)(12 113 165 51)(13 112 166 50)(14 111 167 49)(15 110 168 48)(16 109 127 47)(17 108 128 46)(18 107 129 45)(19 106 130 44)(20 105 131 43)(21 104 132 84)(22 103 133 83)(23 102 134 82)(24 101 135 81)(25 100 136 80)(26 99 137 79)(27 98 138 78)(28 97 139 77)(29 96 140 76)(30 95 141 75)(31 94 142 74)(32 93 143 73)(33 92 144 72)(34 91 145 71)(35 90 146 70)(36 89 147 69)(37 88 148 68)(38 87 149 67)(39 86 150 66)(40 85 151 65)(41 126 152 64)(42 125 153 63)
(1 83 22 62)(2 82 23 61)(3 81 24 60)(4 80 25 59)(5 79 26 58)(6 78 27 57)(7 77 28 56)(8 76 29 55)(9 75 30 54)(10 74 31 53)(11 73 32 52)(12 72 33 51)(13 71 34 50)(14 70 35 49)(15 69 36 48)(16 68 37 47)(17 67 38 46)(18 66 39 45)(19 65 40 44)(20 64 41 43)(21 63 42 84)(85 151 106 130)(86 150 107 129)(87 149 108 128)(88 148 109 127)(89 147 110 168)(90 146 111 167)(91 145 112 166)(92 144 113 165)(93 143 114 164)(94 142 115 163)(95 141 116 162)(96 140 117 161)(97 139 118 160)(98 138 119 159)(99 137 120 158)(100 136 121 157)(101 135 122 156)(102 134 123 155)(103 133 124 154)(104 132 125 153)(105 131 126 152)
G:=sub<Sym(168)| (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), (1,124,154,62)(2,123,155,61)(3,122,156,60)(4,121,157,59)(5,120,158,58)(6,119,159,57)(7,118,160,56)(8,117,161,55)(9,116,162,54)(10,115,163,53)(11,114,164,52)(12,113,165,51)(13,112,166,50)(14,111,167,49)(15,110,168,48)(16,109,127,47)(17,108,128,46)(18,107,129,45)(19,106,130,44)(20,105,131,43)(21,104,132,84)(22,103,133,83)(23,102,134,82)(24,101,135,81)(25,100,136,80)(26,99,137,79)(27,98,138,78)(28,97,139,77)(29,96,140,76)(30,95,141,75)(31,94,142,74)(32,93,143,73)(33,92,144,72)(34,91,145,71)(35,90,146,70)(36,89,147,69)(37,88,148,68)(38,87,149,67)(39,86,150,66)(40,85,151,65)(41,126,152,64)(42,125,153,63), (1,83,22,62)(2,82,23,61)(3,81,24,60)(4,80,25,59)(5,79,26,58)(6,78,27,57)(7,77,28,56)(8,76,29,55)(9,75,30,54)(10,74,31,53)(11,73,32,52)(12,72,33,51)(13,71,34,50)(14,70,35,49)(15,69,36,48)(16,68,37,47)(17,67,38,46)(18,66,39,45)(19,65,40,44)(20,64,41,43)(21,63,42,84)(85,151,106,130)(86,150,107,129)(87,149,108,128)(88,148,109,127)(89,147,110,168)(90,146,111,167)(91,145,112,166)(92,144,113,165)(93,143,114,164)(94,142,115,163)(95,141,116,162)(96,140,117,161)(97,139,118,160)(98,138,119,159)(99,137,120,158)(100,136,121,157)(101,135,122,156)(102,134,123,155)(103,133,124,154)(104,132,125,153)(105,131,126,152)>;
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), (1,124,154,62)(2,123,155,61)(3,122,156,60)(4,121,157,59)(5,120,158,58)(6,119,159,57)(7,118,160,56)(8,117,161,55)(9,116,162,54)(10,115,163,53)(11,114,164,52)(12,113,165,51)(13,112,166,50)(14,111,167,49)(15,110,168,48)(16,109,127,47)(17,108,128,46)(18,107,129,45)(19,106,130,44)(20,105,131,43)(21,104,132,84)(22,103,133,83)(23,102,134,82)(24,101,135,81)(25,100,136,80)(26,99,137,79)(27,98,138,78)(28,97,139,77)(29,96,140,76)(30,95,141,75)(31,94,142,74)(32,93,143,73)(33,92,144,72)(34,91,145,71)(35,90,146,70)(36,89,147,69)(37,88,148,68)(38,87,149,67)(39,86,150,66)(40,85,151,65)(41,126,152,64)(42,125,153,63), (1,83,22,62)(2,82,23,61)(3,81,24,60)(4,80,25,59)(5,79,26,58)(6,78,27,57)(7,77,28,56)(8,76,29,55)(9,75,30,54)(10,74,31,53)(11,73,32,52)(12,72,33,51)(13,71,34,50)(14,70,35,49)(15,69,36,48)(16,68,37,47)(17,67,38,46)(18,66,39,45)(19,65,40,44)(20,64,41,43)(21,63,42,84)(85,151,106,130)(86,150,107,129)(87,149,108,128)(88,148,109,127)(89,147,110,168)(90,146,111,167)(91,145,112,166)(92,144,113,165)(93,143,114,164)(94,142,115,163)(95,141,116,162)(96,140,117,161)(97,139,118,160)(98,138,119,159)(99,137,120,158)(100,136,121,157)(101,135,122,156)(102,134,123,155)(103,133,124,154)(104,132,125,153)(105,131,126,152) );
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)], [(1,124,154,62),(2,123,155,61),(3,122,156,60),(4,121,157,59),(5,120,158,58),(6,119,159,57),(7,118,160,56),(8,117,161,55),(9,116,162,54),(10,115,163,53),(11,114,164,52),(12,113,165,51),(13,112,166,50),(14,111,167,49),(15,110,168,48),(16,109,127,47),(17,108,128,46),(18,107,129,45),(19,106,130,44),(20,105,131,43),(21,104,132,84),(22,103,133,83),(23,102,134,82),(24,101,135,81),(25,100,136,80),(26,99,137,79),(27,98,138,78),(28,97,139,77),(29,96,140,76),(30,95,141,75),(31,94,142,74),(32,93,143,73),(33,92,144,72),(34,91,145,71),(35,90,146,70),(36,89,147,69),(37,88,148,68),(38,87,149,67),(39,86,150,66),(40,85,151,65),(41,126,152,64),(42,125,153,63)], [(1,83,22,62),(2,82,23,61),(3,81,24,60),(4,80,25,59),(5,79,26,58),(6,78,27,57),(7,77,28,56),(8,76,29,55),(9,75,30,54),(10,74,31,53),(11,73,32,52),(12,72,33,51),(13,71,34,50),(14,70,35,49),(15,69,36,48),(16,68,37,47),(17,67,38,46),(18,66,39,45),(19,65,40,44),(20,64,41,43),(21,63,42,84),(85,151,106,130),(86,150,107,129),(87,149,108,128),(88,148,109,127),(89,147,110,168),(90,146,111,167),(91,145,112,166),(92,144,113,165),(93,143,114,164),(94,142,115,163),(95,141,116,162),(96,140,117,161),(97,139,118,160),(98,138,119,159),(99,137,120,158),(100,136,121,157),(101,135,122,156),(102,134,123,155),(103,133,124,154),(104,132,125,153),(105,131,126,152)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | ··· | 6G | 7A | 7B | 7C | 14A | ··· | 14U | 21A | ··· | 21F | 42A | ··· | 42AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | 7 | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 42 | 42 | 42 | 42 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | D7 | C3⋊D4 | Dic7 | D14 | D21 | C7⋊D4 | Dic21 | D42 | C21⋊7D4 |
| kernel | C42.38D4 | C2×Dic21 | C22×C42 | C2×C42 | C22×C14 | C42 | C2×C14 | C2×C14 | C22×C6 | C14 | C2×C6 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 2 | 1 | 3 | 4 | 6 | 3 | 6 | 12 | 12 | 6 | 24 |
Matrix representation of C42.38D4 ►in GL4(𝔽337) generated by
| 332 | 236 | 0 | 0 |
| 202 | 171 | 0 | 0 |
| 0 | 0 | 285 | 0 |
| 0 | 0 | 308 | 162 |
| 5 | 71 | 0 | 0 |
| 180 | 332 | 0 | 0 |
| 0 | 0 | 42 | 213 |
| 0 | 0 | 180 | 295 |
| 5 | 71 | 0 | 0 |
| 180 | 332 | 0 | 0 |
| 0 | 0 | 42 | 213 |
| 0 | 0 | 55 | 295 |
G:=sub<GL(4,GF(337))| [332,202,0,0,236,171,0,0,0,0,285,308,0,0,0,162],[5,180,0,0,71,332,0,0,0,0,42,180,0,0,213,295],[5,180,0,0,71,332,0,0,0,0,42,55,0,0,213,295] >;
C42.38D4 in GAP, Magma, Sage, TeX
C_{42}._{38}D_4 % in TeX
G:=Group("C42.38D4"); // GroupNames label
G:=SmallGroup(336,105);
// by ID
G=gap.SmallGroup(336,105);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,964,10373]);
// Polycyclic
G:=Group<a,b,c|a^42=b^4=1,c^2=a^21,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^21*b^-1>;
// generators/relations