metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (Q8×C10)⋊16C4, (C2×Q8)⋊4Dic5, C20.208(C2×D4), (C2×C20).193D4, Q8.5(C2×Dic5), (C22×Q8).2D5, Q8⋊Dic5⋊37C2, (C2×Q8).166D10, C20.81(C22⋊C4), C20.141(C22×C4), (C2×C20).475C23, (C22×C10).199D4, (C22×C4).154D10, C23.87(C5⋊D4), C5⋊5(C23.38D4), C4.11(C23.D5), C4.12(C22×Dic5), C2.5(C20.C23), C4⋊Dic5.353C22, (Q8×C10).201C22, C10.101(C8.C22), (C22×C20).201C22, C22.21(C23.D5), C23.21D10.19C2, (Q8×C2×C10).2C2, C4.92(C2×C5⋊D4), (C5×Q8).38(C2×C4), (C2×C20).293(C2×C4), (C2×C10).559(C2×D4), (C2×C4).25(C2×Dic5), C22.94(C2×C5⋊D4), C2.14(C2×C23.D5), (C2×C4).198(C5⋊D4), C10.119(C2×C22⋊C4), (C2×C4).561(C22×D5), (C2×C4.Dic5).27C2, (C2×C5⋊2C8).174C22, (C2×C10).179(C22⋊C4), SmallGroup(320,852)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (Q8×C10)⋊16C4
G = < a,b,c,d | a10=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=a5b-1c >
Subgroups: 366 in 150 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, C2×C10, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C23.38D4, C2×C5⋊2C8, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C22×C20, Q8×C10, Q8×C10, Q8⋊Dic5, C2×C4.Dic5, C23.21D10, Q8×C2×C10, (Q8×C10)⋊16C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C8.C22, C2×Dic5, C5⋊D4, C22×D5, C23.38D4, C23.D5, C22×Dic5, C2×C5⋊D4, C20.C23, C2×C23.D5, (Q8×C10)⋊16C4
►On 160 points
Generators in S160
(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 45 16 32)(2 46 17 33)(3 47 18 34)(4 48 19 35)(5 49 20 36)(6 50 11 37)(7 41 12 38)(8 42 13 39)(9 43 14 40)(10 44 15 31)(21 132 145 157)(22 133 146 158)(23 134 147 159)(24 135 148 160)(25 136 149 151)(26 137 150 152)(27 138 141 153)(28 139 142 154)(29 140 143 155)(30 131 144 156)(51 63 90 78)(52 64 81 79)(53 65 82 80)(54 66 83 71)(55 67 84 72)(56 68 85 73)(57 69 86 74)(58 70 87 75)(59 61 88 76)(60 62 89 77)(91 128 116 104)(92 129 117 105)(93 130 118 106)(94 121 119 107)(95 122 120 108)(96 123 111 109)(97 124 112 110)(98 125 113 101)(99 126 114 102)(100 127 115 103)
(1 72 16 67)(2 73 17 68)(3 74 18 69)(4 75 19 70)(5 76 20 61)(6 77 11 62)(7 78 12 63)(8 79 13 64)(9 80 14 65)(10 71 15 66)(21 92 145 117)(22 93 146 118)(23 94 147 119)(24 95 148 120)(25 96 149 111)(26 97 150 112)(27 98 141 113)(28 99 142 114)(29 100 143 115)(30 91 144 116)(31 54 44 83)(32 55 45 84)(33 56 46 85)(34 57 47 86)(35 58 48 87)(36 59 49 88)(37 60 50 89)(38 51 41 90)(39 52 42 81)(40 53 43 82)(101 153 125 138)(102 154 126 139)(103 155 127 140)(104 156 128 131)(105 157 129 132)(106 158 130 133)(107 159 121 134)(108 160 122 135)(109 151 123 136)(110 152 124 137)
(1 93 11 98)(2 117 12 112)(3 91 13 96)(4 115 14 120)(5 99 15 94)(6 113 16 118)(7 97 17 92)(8 111 18 116)(9 95 19 100)(10 119 20 114)(21 85 26 51)(22 55 27 89)(23 83 28 59)(24 53 29 87)(25 81 30 57)(31 121 49 126)(32 106 50 101)(33 129 41 124)(34 104 42 109)(35 127 43 122)(36 102 44 107)(37 125 45 130)(38 110 46 105)(39 123 47 128)(40 108 48 103)(52 144 86 149)(54 142 88 147)(56 150 90 145)(58 148 82 143)(60 146 84 141)(61 134 71 139)(62 158 72 153)(63 132 73 137)(64 156 74 151)(65 140 75 135)(66 154 76 159)(67 138 77 133)(68 152 78 157)(69 136 79 131)(70 160 80 155)
G:=sub<Sym(160)| (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), (1,45,16,32)(2,46,17,33)(3,47,18,34)(4,48,19,35)(5,49,20,36)(6,50,11,37)(7,41,12,38)(8,42,13,39)(9,43,14,40)(10,44,15,31)(21,132,145,157)(22,133,146,158)(23,134,147,159)(24,135,148,160)(25,136,149,151)(26,137,150,152)(27,138,141,153)(28,139,142,154)(29,140,143,155)(30,131,144,156)(51,63,90,78)(52,64,81,79)(53,65,82,80)(54,66,83,71)(55,67,84,72)(56,68,85,73)(57,69,86,74)(58,70,87,75)(59,61,88,76)(60,62,89,77)(91,128,116,104)(92,129,117,105)(93,130,118,106)(94,121,119,107)(95,122,120,108)(96,123,111,109)(97,124,112,110)(98,125,113,101)(99,126,114,102)(100,127,115,103), (1,72,16,67)(2,73,17,68)(3,74,18,69)(4,75,19,70)(5,76,20,61)(6,77,11,62)(7,78,12,63)(8,79,13,64)(9,80,14,65)(10,71,15,66)(21,92,145,117)(22,93,146,118)(23,94,147,119)(24,95,148,120)(25,96,149,111)(26,97,150,112)(27,98,141,113)(28,99,142,114)(29,100,143,115)(30,91,144,116)(31,54,44,83)(32,55,45,84)(33,56,46,85)(34,57,47,86)(35,58,48,87)(36,59,49,88)(37,60,50,89)(38,51,41,90)(39,52,42,81)(40,53,43,82)(101,153,125,138)(102,154,126,139)(103,155,127,140)(104,156,128,131)(105,157,129,132)(106,158,130,133)(107,159,121,134)(108,160,122,135)(109,151,123,136)(110,152,124,137), (1,93,11,98)(2,117,12,112)(3,91,13,96)(4,115,14,120)(5,99,15,94)(6,113,16,118)(7,97,17,92)(8,111,18,116)(9,95,19,100)(10,119,20,114)(21,85,26,51)(22,55,27,89)(23,83,28,59)(24,53,29,87)(25,81,30,57)(31,121,49,126)(32,106,50,101)(33,129,41,124)(34,104,42,109)(35,127,43,122)(36,102,44,107)(37,125,45,130)(38,110,46,105)(39,123,47,128)(40,108,48,103)(52,144,86,149)(54,142,88,147)(56,150,90,145)(58,148,82,143)(60,146,84,141)(61,134,71,139)(62,158,72,153)(63,132,73,137)(64,156,74,151)(65,140,75,135)(66,154,76,159)(67,138,77,133)(68,152,78,157)(69,136,79,131)(70,160,80,155)>;
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), (1,45,16,32)(2,46,17,33)(3,47,18,34)(4,48,19,35)(5,49,20,36)(6,50,11,37)(7,41,12,38)(8,42,13,39)(9,43,14,40)(10,44,15,31)(21,132,145,157)(22,133,146,158)(23,134,147,159)(24,135,148,160)(25,136,149,151)(26,137,150,152)(27,138,141,153)(28,139,142,154)(29,140,143,155)(30,131,144,156)(51,63,90,78)(52,64,81,79)(53,65,82,80)(54,66,83,71)(55,67,84,72)(56,68,85,73)(57,69,86,74)(58,70,87,75)(59,61,88,76)(60,62,89,77)(91,128,116,104)(92,129,117,105)(93,130,118,106)(94,121,119,107)(95,122,120,108)(96,123,111,109)(97,124,112,110)(98,125,113,101)(99,126,114,102)(100,127,115,103), (1,72,16,67)(2,73,17,68)(3,74,18,69)(4,75,19,70)(5,76,20,61)(6,77,11,62)(7,78,12,63)(8,79,13,64)(9,80,14,65)(10,71,15,66)(21,92,145,117)(22,93,146,118)(23,94,147,119)(24,95,148,120)(25,96,149,111)(26,97,150,112)(27,98,141,113)(28,99,142,114)(29,100,143,115)(30,91,144,116)(31,54,44,83)(32,55,45,84)(33,56,46,85)(34,57,47,86)(35,58,48,87)(36,59,49,88)(37,60,50,89)(38,51,41,90)(39,52,42,81)(40,53,43,82)(101,153,125,138)(102,154,126,139)(103,155,127,140)(104,156,128,131)(105,157,129,132)(106,158,130,133)(107,159,121,134)(108,160,122,135)(109,151,123,136)(110,152,124,137), (1,93,11,98)(2,117,12,112)(3,91,13,96)(4,115,14,120)(5,99,15,94)(6,113,16,118)(7,97,17,92)(8,111,18,116)(9,95,19,100)(10,119,20,114)(21,85,26,51)(22,55,27,89)(23,83,28,59)(24,53,29,87)(25,81,30,57)(31,121,49,126)(32,106,50,101)(33,129,41,124)(34,104,42,109)(35,127,43,122)(36,102,44,107)(37,125,45,130)(38,110,46,105)(39,123,47,128)(40,108,48,103)(52,144,86,149)(54,142,88,147)(56,150,90,145)(58,148,82,143)(60,146,84,141)(61,134,71,139)(62,158,72,153)(63,132,73,137)(64,156,74,151)(65,140,75,135)(66,154,76,159)(67,138,77,133)(68,152,78,157)(69,136,79,131)(70,160,80,155) );
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)], [(1,45,16,32),(2,46,17,33),(3,47,18,34),(4,48,19,35),(5,49,20,36),(6,50,11,37),(7,41,12,38),(8,42,13,39),(9,43,14,40),(10,44,15,31),(21,132,145,157),(22,133,146,158),(23,134,147,159),(24,135,148,160),(25,136,149,151),(26,137,150,152),(27,138,141,153),(28,139,142,154),(29,140,143,155),(30,131,144,156),(51,63,90,78),(52,64,81,79),(53,65,82,80),(54,66,83,71),(55,67,84,72),(56,68,85,73),(57,69,86,74),(58,70,87,75),(59,61,88,76),(60,62,89,77),(91,128,116,104),(92,129,117,105),(93,130,118,106),(94,121,119,107),(95,122,120,108),(96,123,111,109),(97,124,112,110),(98,125,113,101),(99,126,114,102),(100,127,115,103)], [(1,72,16,67),(2,73,17,68),(3,74,18,69),(4,75,19,70),(5,76,20,61),(6,77,11,62),(7,78,12,63),(8,79,13,64),(9,80,14,65),(10,71,15,66),(21,92,145,117),(22,93,146,118),(23,94,147,119),(24,95,148,120),(25,96,149,111),(26,97,150,112),(27,98,141,113),(28,99,142,114),(29,100,143,115),(30,91,144,116),(31,54,44,83),(32,55,45,84),(33,56,46,85),(34,57,47,86),(35,58,48,87),(36,59,49,88),(37,60,50,89),(38,51,41,90),(39,52,42,81),(40,53,43,82),(101,153,125,138),(102,154,126,139),(103,155,127,140),(104,156,128,131),(105,157,129,132),(106,158,130,133),(107,159,121,134),(108,160,122,135),(109,151,123,136),(110,152,124,137)], [(1,93,11,98),(2,117,12,112),(3,91,13,96),(4,115,14,120),(5,99,15,94),(6,113,16,118),(7,97,17,92),(8,111,18,116),(9,95,19,100),(10,119,20,114),(21,85,26,51),(22,55,27,89),(23,83,28,59),(24,53,29,87),(25,81,30,57),(31,121,49,126),(32,106,50,101),(33,129,41,124),(34,104,42,109),(35,127,43,122),(36,102,44,107),(37,125,45,130),(38,110,46,105),(39,123,47,128),(40,108,48,103),(52,144,86,149),(54,142,88,147),(56,150,90,145),(58,148,82,143),(60,146,84,141),(61,134,71,139),(62,158,72,153),(63,132,73,137),(64,156,74,151),(65,140,75,135),(66,154,76,159),(67,138,77,133),(68,152,78,157),(69,136,79,131),(70,160,80,155)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | Dic5 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | C20.C23 |
| kernel | (Q8×C10)⋊16C4 | Q8⋊Dic5 | C2×C4.Dic5 | C23.21D10 | Q8×C2×C10 | Q8×C10 | C2×C20 | C22×C10 | C22×Q8 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 2 | 8 | 4 | 12 | 4 | 2 | 8 |
Matrix representation of (Q8×C10)⋊16C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 38 | 0 | 10 | 0 |
| 0 | 0 | 0 | 38 | 0 | 10 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 33 | 22 | 0 | 0 | 0 | 0 |
| 40 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 27 | 0 | 0 |
| 0 | 0 | 27 | 34 | 0 | 0 |
| 0 | 0 | 17 | 10 | 14 | 7 |
| 0 | 0 | 10 | 24 | 7 | 27 |
| 33 | 22 | 0 | 0 | 0 | 0 |
| 25 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 39 | 0 |
| 0 | 0 | 0 | 1 | 0 | 39 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,38,0,0,0,0,4,0,38,0,0,0,0,10,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[33,40,0,0,0,0,22,8,0,0,0,0,0,0,7,27,17,10,0,0,27,34,10,24,0,0,0,0,14,7,0,0,0,0,7,27],[33,25,0,0,0,0,22,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,39,0,40,0,0,0,0,39,0,40] >;
(Q8×C10)⋊16C4 in GAP, Magma, Sage, TeX
(Q_8\times C_{10})\rtimes_{16}C_4 % in TeX
G:=Group("(Q8xC10):16C4"); // GroupNames label
G:=SmallGroup(320,852);
// by ID
G=gap.SmallGroup(320,852);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^5*b^-1*c>;
// generators/relations