metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊C4⋊7D5, C40⋊5C4⋊11C2, (C2×C8).19D10, C4⋊C4.153D10, Q8⋊Dic5⋊9C2, (C2×Q8).20D10, D10⋊1C8.4C2, C4.57(C4○D20), C10.71(C4○D8), (C2×C40).19C22, D10⋊3Q8.5C2, C20.Q8⋊12C2, (C22×D5).28D4, C22.204(D4×D5), C20.163(C4○D4), C4.88(D4⋊2D5), (C2×C20).254C23, (C2×Dic5).212D4, C5⋊2(C23.20D4), C4⋊Dic5.98C22, (Q8×C10).37C22, C2.10(Q8.D10), C2.18(SD16⋊D5), C10.36(C8.C22), C2.18(D10.12D4), C10.26(C22.D4), C4⋊C4⋊7D5.2C2, (C5×Q8⋊C4)⋊7C2, (C2×C4×D5).30C22, (C2×C10).267(C2×D4), (C5×C4⋊C4).55C22, (C2×C5⋊2C8).44C22, (C2×C4).361(C22×D5), SmallGroup(320,441)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C8).D10
G = < a,b,c,d | a2=b8=1, c10=a, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=b-1, dcd-1=b4c9 >
Subgroups: 366 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.20D4, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, Q8×C10, C20.Q8, C40⋊5C4, D10⋊1C8, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C4⋊7D5, D10⋊3Q8, (C2×C8).D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8.C22, C22×D5, C23.20D4, C4○D20, D4×D5, D4⋊2D5, D10.12D4, SD16⋊D5, Q8.D10, (C2×C8).D10
►On 160 points
Generators in S160
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 70 36 88 105 159 46 122)(2 99 47 61 106 133 37 150)(3 72 38 90 107 141 48 124)(4 81 49 63 108 135 39 152)(5 74 40 92 109 143 50 126)(6 83 51 65 110 137 21 154)(7 76 22 94 111 145 52 128)(8 85 53 67 112 139 23 156)(9 78 24 96 113 147 54 130)(10 87 55 69 114 121 25 158)(11 80 26 98 115 149 56 132)(12 89 57 71 116 123 27 160)(13 62 28 100 117 151 58 134)(14 91 59 73 118 125 29 142)(15 64 30 82 119 153 60 136)(16 93 41 75 120 127 31 144)(17 66 32 84 101 155 42 138)(18 95 43 77 102 129 33 146)(19 68 34 86 103 157 44 140)(20 97 45 79 104 131 35 148)
(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 10 115 104)(2 103 116 9)(3 8 117 102)(4 101 118 7)(5 6 119 120)(11 20 105 114)(12 113 106 19)(13 18 107 112)(14 111 108 17)(15 16 109 110)(21 60 41 40)(22 39 42 59)(23 58 43 38)(24 37 44 57)(25 56 45 36)(26 35 46 55)(27 54 47 34)(28 33 48 53)(29 52 49 32)(30 31 50 51)(61 68 160 147)(62 146 141 67)(63 66 142 145)(64 144 143 65)(69 80 148 159)(70 158 149 79)(71 78 150 157)(72 156 151 77)(73 76 152 155)(74 154 153 75)(81 84 125 128)(82 127 126 83)(85 100 129 124)(86 123 130 99)(87 98 131 122)(88 121 132 97)(89 96 133 140)(90 139 134 95)(91 94 135 138)(92 137 136 93)
G:=sub<Sym(160)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,70,36,88,105,159,46,122)(2,99,47,61,106,133,37,150)(3,72,38,90,107,141,48,124)(4,81,49,63,108,135,39,152)(5,74,40,92,109,143,50,126)(6,83,51,65,110,137,21,154)(7,76,22,94,111,145,52,128)(8,85,53,67,112,139,23,156)(9,78,24,96,113,147,54,130)(10,87,55,69,114,121,25,158)(11,80,26,98,115,149,56,132)(12,89,57,71,116,123,27,160)(13,62,28,100,117,151,58,134)(14,91,59,73,118,125,29,142)(15,64,30,82,119,153,60,136)(16,93,41,75,120,127,31,144)(17,66,32,84,101,155,42,138)(18,95,43,77,102,129,33,146)(19,68,34,86,103,157,44,140)(20,97,45,79,104,131,35,148), (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), (1,10,115,104)(2,103,116,9)(3,8,117,102)(4,101,118,7)(5,6,119,120)(11,20,105,114)(12,113,106,19)(13,18,107,112)(14,111,108,17)(15,16,109,110)(21,60,41,40)(22,39,42,59)(23,58,43,38)(24,37,44,57)(25,56,45,36)(26,35,46,55)(27,54,47,34)(28,33,48,53)(29,52,49,32)(30,31,50,51)(61,68,160,147)(62,146,141,67)(63,66,142,145)(64,144,143,65)(69,80,148,159)(70,158,149,79)(71,78,150,157)(72,156,151,77)(73,76,152,155)(74,154,153,75)(81,84,125,128)(82,127,126,83)(85,100,129,124)(86,123,130,99)(87,98,131,122)(88,121,132,97)(89,96,133,140)(90,139,134,95)(91,94,135,138)(92,137,136,93)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,70,36,88,105,159,46,122)(2,99,47,61,106,133,37,150)(3,72,38,90,107,141,48,124)(4,81,49,63,108,135,39,152)(5,74,40,92,109,143,50,126)(6,83,51,65,110,137,21,154)(7,76,22,94,111,145,52,128)(8,85,53,67,112,139,23,156)(9,78,24,96,113,147,54,130)(10,87,55,69,114,121,25,158)(11,80,26,98,115,149,56,132)(12,89,57,71,116,123,27,160)(13,62,28,100,117,151,58,134)(14,91,59,73,118,125,29,142)(15,64,30,82,119,153,60,136)(16,93,41,75,120,127,31,144)(17,66,32,84,101,155,42,138)(18,95,43,77,102,129,33,146)(19,68,34,86,103,157,44,140)(20,97,45,79,104,131,35,148), (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), (1,10,115,104)(2,103,116,9)(3,8,117,102)(4,101,118,7)(5,6,119,120)(11,20,105,114)(12,113,106,19)(13,18,107,112)(14,111,108,17)(15,16,109,110)(21,60,41,40)(22,39,42,59)(23,58,43,38)(24,37,44,57)(25,56,45,36)(26,35,46,55)(27,54,47,34)(28,33,48,53)(29,52,49,32)(30,31,50,51)(61,68,160,147)(62,146,141,67)(63,66,142,145)(64,144,143,65)(69,80,148,159)(70,158,149,79)(71,78,150,157)(72,156,151,77)(73,76,152,155)(74,154,153,75)(81,84,125,128)(82,127,126,83)(85,100,129,124)(86,123,130,99)(87,98,131,122)(88,121,132,97)(89,96,133,140)(90,139,134,95)(91,94,135,138)(92,137,136,93) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,70,36,88,105,159,46,122),(2,99,47,61,106,133,37,150),(3,72,38,90,107,141,48,124),(4,81,49,63,108,135,39,152),(5,74,40,92,109,143,50,126),(6,83,51,65,110,137,21,154),(7,76,22,94,111,145,52,128),(8,85,53,67,112,139,23,156),(9,78,24,96,113,147,54,130),(10,87,55,69,114,121,25,158),(11,80,26,98,115,149,56,132),(12,89,57,71,116,123,27,160),(13,62,28,100,117,151,58,134),(14,91,59,73,118,125,29,142),(15,64,30,82,119,153,60,136),(16,93,41,75,120,127,31,144),(17,66,32,84,101,155,42,138),(18,95,43,77,102,129,33,146),(19,68,34,86,103,157,44,140),(20,97,45,79,104,131,35,148)], [(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)], [(1,10,115,104),(2,103,116,9),(3,8,117,102),(4,101,118,7),(5,6,119,120),(11,20,105,114),(12,113,106,19),(13,18,107,112),(14,111,108,17),(15,16,109,110),(21,60,41,40),(22,39,42,59),(23,58,43,38),(24,37,44,57),(25,56,45,36),(26,35,46,55),(27,54,47,34),(28,33,48,53),(29,52,49,32),(30,31,50,51),(61,68,160,147),(62,146,141,67),(63,66,142,145),(64,144,143,65),(69,80,148,159),(70,158,149,79),(71,78,150,157),(72,156,151,77),(73,76,152,155),(74,154,153,75),(81,84,125,128),(82,127,126,83),(85,100,129,124),(86,123,130,99),(87,98,131,122),(88,121,132,97),(89,96,133,140),(90,139,134,95),(91,94,135,138),(92,137,136,93)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8.C22 | D4⋊2D5 | D4×D5 | SD16⋊D5 | Q8.D10 |
| kernel | (C2×C8).D10 | C20.Q8 | C40⋊5C4 | D10⋊1C8 | Q8⋊Dic5 | C5×Q8⋊C4 | C4⋊C4⋊7D5 | D10⋊3Q8 | C2×Dic5 | C22×D5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of (C2×C8).D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 2 | 28 | 0 | 0 |
| 13 | 39 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 3 |
| 32 | 28 | 0 | 0 |
| 13 | 28 | 0 | 0 |
| 0 | 0 | 0 | 38 |
| 0 | 0 | 14 | 0 |
| 28 | 32 | 0 | 0 |
| 28 | 13 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 14 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[2,13,0,0,28,39,0,0,0,0,14,0,0,0,0,3],[32,13,0,0,28,28,0,0,0,0,0,14,0,0,38,0],[28,28,0,0,32,13,0,0,0,0,0,14,0,0,3,0] >;
(C2×C8).D10 in GAP, Magma, Sage, TeX
(C_2\times C_8).D_{10} % in TeX
G:=Group("(C2xC8).D10"); // GroupNames label
G:=SmallGroup(320,441);
// by ID
G=gap.SmallGroup(320,441);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,184,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=1,c^10=a,d^2=a*b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^9>;
// generators/relations