metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.138D10, C10.712+ (1+4), C4.4D4⋊7D5, (C2×Q8).82D10, D10⋊D4⋊38C2, (C2×D4).108D10, C42⋊D5⋊35C2, C22⋊C4.72D10, Dic5⋊D4⋊33C2, Dic5⋊Q8⋊21C2, Dic5⋊4D4⋊29C2, C20.23D4⋊19C2, (C2×C10).214C24, (C2×C20).630C23, (C4×C20).239C22, C2.73(D4⋊6D10), C23.36(C22×D5), Dic5.43(C4○D4), Dic5.5D4⋊38C2, (D4×C10).208C22, (C2×D20).167C22, (C22×C10).44C23, (Q8×C10).123C22, (C22×D5).94C23, C22.235(C23×D5), C23.D5.51C22, C23.11D10⋊17C2, C5⋊4(C22.49C24), (C2×Dic5).261C23, (C4×Dic5).138C22, D10⋊C4.134C22, (C2×Dic10).181C22, C10.D4.141C22, (C22×Dic5).139C22, C2.73(D5×C4○D4), (C5×C4.4D4)⋊8C2, C10.185(C2×C4○D4), (C2×C4×D5).265C22, (C2×C4).73(C22×D5), (C2×C5⋊D4).57C22, (C5×C22⋊C4).61C22, SmallGroup(320,1342)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 854 in 236 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×13], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×8], Q8 [×2], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×4], Dic5 [×4], C20 [×5], D10 [×6], C2×C10, C2×C10 [×6], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4, C4.4D4 [×3], C4⋊Q8, Dic10, C4×D5 [×4], D20, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C5×D4, C5×Q8, C22×D5 [×2], C22×C10 [×2], C22.49C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×6], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C42⋊D5 [×2], C23.11D10 [×2], Dic5⋊4D4 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], Dic5⋊D4 [×2], Dic5⋊Q8, C20.23D4, C5×C4.4D4, C42.138D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.49C24, C23×D5, D4⋊6D10, D5×C4○D4 [×2], C42.138D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >
►On 160 points
Generators in S160
(1 50 30 113)(2 71 21 144)(3 42 22 115)(4 73 23 146)(5 44 24 117)(6 75 25 148)(7 46 26 119)(8 77 27 150)(9 48 28 111)(10 79 29 142)(11 101 93 38)(12 132 94 69)(13 103 95 40)(14 134 96 61)(15 105 97 32)(16 136 98 63)(17 107 99 34)(18 138 100 65)(19 109 91 36)(20 140 92 67)(31 154 104 87)(33 156 106 89)(35 158 108 81)(37 160 110 83)(39 152 102 85)(41 51 114 124)(43 53 116 126)(45 55 118 128)(47 57 120 130)(49 59 112 122)(52 145 125 72)(54 147 127 74)(56 149 129 76)(58 141 121 78)(60 143 123 80)(62 155 135 88)(64 157 137 90)(66 159 139 82)(68 151 131 84)(70 153 133 86)
(1 133 123 103)(2 61 124 31)(3 135 125 105)(4 63 126 33)(5 137 127 107)(6 65 128 35)(7 139 129 109)(8 67 130 37)(9 131 121 101)(10 69 122 39)(11 111 151 141)(12 49 152 79)(13 113 153 143)(14 41 154 71)(15 115 155 145)(16 43 156 73)(17 117 157 147)(18 45 158 75)(19 119 159 149)(20 47 160 77)(21 134 51 104)(22 62 52 32)(23 136 53 106)(24 64 54 34)(25 138 55 108)(26 66 56 36)(27 140 57 110)(28 68 58 38)(29 132 59 102)(30 70 60 40)(42 88 72 97)(44 90 74 99)(46 82 76 91)(48 84 78 93)(50 86 80 95)(81 148 100 118)(83 150 92 120)(85 142 94 112)(87 144 96 114)(89 146 98 116)
(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)
(1 102 123 132)(2 101 124 131)(3 110 125 140)(4 109 126 139)(5 108 127 138)(6 107 128 137)(7 106 129 136)(8 105 130 135)(9 104 121 134)(10 103 122 133)(11 114 151 144)(12 113 152 143)(13 112 153 142)(14 111 154 141)(15 120 155 150)(16 119 156 149)(17 118 157 148)(18 117 158 147)(19 116 159 146)(20 115 160 145)(21 38 51 68)(22 37 52 67)(23 36 53 66)(24 35 54 65)(25 34 55 64)(26 33 56 63)(27 32 57 62)(28 31 58 61)(29 40 59 70)(30 39 60 69)(41 84 71 93)(42 83 72 92)(43 82 73 91)(44 81 74 100)(45 90 75 99)(46 89 76 98)(47 88 77 97)(48 87 78 96)(49 86 79 95)(50 85 80 94)
G:=sub<Sym(160)| (1,50,30,113)(2,71,21,144)(3,42,22,115)(4,73,23,146)(5,44,24,117)(6,75,25,148)(7,46,26,119)(8,77,27,150)(9,48,28,111)(10,79,29,142)(11,101,93,38)(12,132,94,69)(13,103,95,40)(14,134,96,61)(15,105,97,32)(16,136,98,63)(17,107,99,34)(18,138,100,65)(19,109,91,36)(20,140,92,67)(31,154,104,87)(33,156,106,89)(35,158,108,81)(37,160,110,83)(39,152,102,85)(41,51,114,124)(43,53,116,126)(45,55,118,128)(47,57,120,130)(49,59,112,122)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,103)(2,61,124,31)(3,135,125,105)(4,63,126,33)(5,137,127,107)(6,65,128,35)(7,139,129,109)(8,67,130,37)(9,131,121,101)(10,69,122,39)(11,111,151,141)(12,49,152,79)(13,113,153,143)(14,41,154,71)(15,115,155,145)(16,43,156,73)(17,117,157,147)(18,45,158,75)(19,119,159,149)(20,47,160,77)(21,134,51,104)(22,62,52,32)(23,136,53,106)(24,64,54,34)(25,138,55,108)(26,66,56,36)(27,140,57,110)(28,68,58,38)(29,132,59,102)(30,70,60,40)(42,88,72,97)(44,90,74,99)(46,82,76,91)(48,84,78,93)(50,86,80,95)(81,148,100,118)(83,150,92,120)(85,142,94,112)(87,144,96,114)(89,146,98,116), (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), (1,102,123,132)(2,101,124,131)(3,110,125,140)(4,109,126,139)(5,108,127,138)(6,107,128,137)(7,106,129,136)(8,105,130,135)(9,104,121,134)(10,103,122,133)(11,114,151,144)(12,113,152,143)(13,112,153,142)(14,111,154,141)(15,120,155,150)(16,119,156,149)(17,118,157,148)(18,117,158,147)(19,116,159,146)(20,115,160,145)(21,38,51,68)(22,37,52,67)(23,36,53,66)(24,35,54,65)(25,34,55,64)(26,33,56,63)(27,32,57,62)(28,31,58,61)(29,40,59,70)(30,39,60,69)(41,84,71,93)(42,83,72,92)(43,82,73,91)(44,81,74,100)(45,90,75,99)(46,89,76,98)(47,88,77,97)(48,87,78,96)(49,86,79,95)(50,85,80,94)>;
G:=Group( (1,50,30,113)(2,71,21,144)(3,42,22,115)(4,73,23,146)(5,44,24,117)(6,75,25,148)(7,46,26,119)(8,77,27,150)(9,48,28,111)(10,79,29,142)(11,101,93,38)(12,132,94,69)(13,103,95,40)(14,134,96,61)(15,105,97,32)(16,136,98,63)(17,107,99,34)(18,138,100,65)(19,109,91,36)(20,140,92,67)(31,154,104,87)(33,156,106,89)(35,158,108,81)(37,160,110,83)(39,152,102,85)(41,51,114,124)(43,53,116,126)(45,55,118,128)(47,57,120,130)(49,59,112,122)(52,145,125,72)(54,147,127,74)(56,149,129,76)(58,141,121,78)(60,143,123,80)(62,155,135,88)(64,157,137,90)(66,159,139,82)(68,151,131,84)(70,153,133,86), (1,133,123,103)(2,61,124,31)(3,135,125,105)(4,63,126,33)(5,137,127,107)(6,65,128,35)(7,139,129,109)(8,67,130,37)(9,131,121,101)(10,69,122,39)(11,111,151,141)(12,49,152,79)(13,113,153,143)(14,41,154,71)(15,115,155,145)(16,43,156,73)(17,117,157,147)(18,45,158,75)(19,119,159,149)(20,47,160,77)(21,134,51,104)(22,62,52,32)(23,136,53,106)(24,64,54,34)(25,138,55,108)(26,66,56,36)(27,140,57,110)(28,68,58,38)(29,132,59,102)(30,70,60,40)(42,88,72,97)(44,90,74,99)(46,82,76,91)(48,84,78,93)(50,86,80,95)(81,148,100,118)(83,150,92,120)(85,142,94,112)(87,144,96,114)(89,146,98,116), (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), (1,102,123,132)(2,101,124,131)(3,110,125,140)(4,109,126,139)(5,108,127,138)(6,107,128,137)(7,106,129,136)(8,105,130,135)(9,104,121,134)(10,103,122,133)(11,114,151,144)(12,113,152,143)(13,112,153,142)(14,111,154,141)(15,120,155,150)(16,119,156,149)(17,118,157,148)(18,117,158,147)(19,116,159,146)(20,115,160,145)(21,38,51,68)(22,37,52,67)(23,36,53,66)(24,35,54,65)(25,34,55,64)(26,33,56,63)(27,32,57,62)(28,31,58,61)(29,40,59,70)(30,39,60,69)(41,84,71,93)(42,83,72,92)(43,82,73,91)(44,81,74,100)(45,90,75,99)(46,89,76,98)(47,88,77,97)(48,87,78,96)(49,86,79,95)(50,85,80,94) );
G=PermutationGroup([(1,50,30,113),(2,71,21,144),(3,42,22,115),(4,73,23,146),(5,44,24,117),(6,75,25,148),(7,46,26,119),(8,77,27,150),(9,48,28,111),(10,79,29,142),(11,101,93,38),(12,132,94,69),(13,103,95,40),(14,134,96,61),(15,105,97,32),(16,136,98,63),(17,107,99,34),(18,138,100,65),(19,109,91,36),(20,140,92,67),(31,154,104,87),(33,156,106,89),(35,158,108,81),(37,160,110,83),(39,152,102,85),(41,51,114,124),(43,53,116,126),(45,55,118,128),(47,57,120,130),(49,59,112,122),(52,145,125,72),(54,147,127,74),(56,149,129,76),(58,141,121,78),(60,143,123,80),(62,155,135,88),(64,157,137,90),(66,159,139,82),(68,151,131,84),(70,153,133,86)], [(1,133,123,103),(2,61,124,31),(3,135,125,105),(4,63,126,33),(5,137,127,107),(6,65,128,35),(7,139,129,109),(8,67,130,37),(9,131,121,101),(10,69,122,39),(11,111,151,141),(12,49,152,79),(13,113,153,143),(14,41,154,71),(15,115,155,145),(16,43,156,73),(17,117,157,147),(18,45,158,75),(19,119,159,149),(20,47,160,77),(21,134,51,104),(22,62,52,32),(23,136,53,106),(24,64,54,34),(25,138,55,108),(26,66,56,36),(27,140,57,110),(28,68,58,38),(29,132,59,102),(30,70,60,40),(42,88,72,97),(44,90,74,99),(46,82,76,91),(48,84,78,93),(50,86,80,95),(81,148,100,118),(83,150,92,120),(85,142,94,112),(87,144,96,114),(89,146,98,116)], [(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)], [(1,102,123,132),(2,101,124,131),(3,110,125,140),(4,109,126,139),(5,108,127,138),(6,107,128,137),(7,106,129,136),(8,105,130,135),(9,104,121,134),(10,103,122,133),(11,114,151,144),(12,113,152,143),(13,112,153,142),(14,111,154,141),(15,120,155,150),(16,119,156,149),(17,118,157,148),(18,117,158,147),(19,116,159,146),(20,115,160,145),(21,38,51,68),(22,37,52,67),(23,36,53,66),(24,35,54,65),(25,34,55,64),(26,33,56,63),(27,32,57,62),(28,31,58,61),(29,40,59,70),(30,39,60,69),(41,84,71,93),(42,83,72,92),(43,82,73,91),(44,81,74,100),(45,90,75,99),(46,89,76,98),(47,88,77,97),(48,87,78,96),(49,86,79,95),(50,85,80,94)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 8 |
| 0 | 0 | 0 | 0 | 13 | 26 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 25 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 27 | 40 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 25 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,13,0,0,0,0,8,26],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[9,25,0,0,0,0,5,32,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,1,27,0,0,0,0,0,40],[9,25,0,0,0,0,5,32,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,32,0,0,0,0,0,0,32] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 |
| kernel | C42.138D10 | C42⋊D5 | C23.11D10 | Dic5⋊4D4 | D10⋊D4 | Dic5.5D4 | Dic5⋊D4 | Dic5⋊Q8 | C20.23D4 | C5×C4.4D4 | C4.4D4 | Dic5 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 2 | 8 | 2 | 2 | 1 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{138}D_{10} % in TeX
G:=Group("C4^2.138D10"); // GroupNames label
G:=SmallGroup(320,1342);
// by ID
G=gap.SmallGroup(320,1342);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,387,100,346,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀