metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.22D10, Q8⋊C4⋊1D5, (C8×Dic5)⋊22C2, Q8⋊Dic5⋊2C2, (C2×C8).207D10, (C2×Q8).14D10, C4.Dic10⋊5C2, D20⋊5C4.8C2, D20⋊6C4.3C2, C4.31(C4○D20), C10.68(C4○D8), C20.18(C4○D4), C22.194(D4×D5), C4.57(D4⋊2D5), C2.7(Q8.D10), (C2×C40).195C22, (C2×C20).240C23, (C2×Dic5).137D4, C20.23D4.6C2, (C2×D20).63C22, C4⋊Dic5.88C22, (Q8×C10).23C22, C10.29(C4.4D4), C2.16(SD16⋊3D5), C5⋊3(C42.78C22), (C4×Dic5).258C22, C2.19(Dic5.5D4), (C5×Q8⋊C4)⋊17C2, (C2×C10).253(C2×D4), (C5×C4⋊C4).41C22, (C2×C4).347(C22×D5), (C2×C5⋊2C8).225C22, SmallGroup(320,427)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊Dic5⋊C2
G = < a,b,c,d | a2=b8=1, c10=a, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=ab3, dcd-1=ab4c9 >
Subgroups: 414 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4.4D4, C42.C2, C5⋊2C8, C40, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C42.78C22, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×D20, Q8×C10, D20⋊6C4, C8×Dic5, D20⋊5C4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, C20.23D4, Q8⋊Dic5⋊C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C4○D8, C22×D5, C42.78C22, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, SD16⋊3D5, Q8.D10, Q8⋊Dic5⋊C2
►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 56 124 147 101 34 68 84)(2 158 69 47 102 95 125 25)(3 58 126 149 103 36 70 86)(4 160 71 49 104 97 127 27)(5 60 128 151 105 38 72 88)(6 142 73 51 106 99 129 29)(7 42 130 153 107 40 74 90)(8 144 75 53 108 81 131 31)(9 44 132 155 109 22 76 92)(10 146 77 55 110 83 133 33)(11 46 134 157 111 24 78 94)(12 148 79 57 112 85 135 35)(13 48 136 159 113 26 80 96)(14 150 61 59 114 87 137 37)(15 50 138 141 115 28 62 98)(16 152 63 41 116 89 139 39)(17 52 140 143 117 30 64 100)(18 154 65 43 118 91 121 21)(19 54 122 145 119 32 66 82)(20 156 67 45 120 93 123 23)
(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 98 111 151)(2 160 112 87)(3 96 113 149)(4 158 114 85)(5 94 115 147)(6 156 116 83)(7 92 117 145)(8 154 118 81)(9 90 119 143)(10 152 120 99)(11 88 101 141)(12 150 102 97)(13 86 103 159)(14 148 104 95)(15 84 105 157)(16 146 106 93)(17 82 107 155)(18 144 108 91)(19 100 109 153)(20 142 110 89)(21 121 53 75)(22 64 54 130)(23 139 55 73)(24 62 56 128)(25 137 57 71)(26 80 58 126)(27 135 59 69)(28 78 60 124)(29 133 41 67)(30 76 42 122)(31 131 43 65)(32 74 44 140)(33 129 45 63)(34 72 46 138)(35 127 47 61)(36 70 48 136)(37 125 49 79)(38 68 50 134)(39 123 51 77)(40 66 52 132)
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), (1,56,124,147,101,34,68,84)(2,158,69,47,102,95,125,25)(3,58,126,149,103,36,70,86)(4,160,71,49,104,97,127,27)(5,60,128,151,105,38,72,88)(6,142,73,51,106,99,129,29)(7,42,130,153,107,40,74,90)(8,144,75,53,108,81,131,31)(9,44,132,155,109,22,76,92)(10,146,77,55,110,83,133,33)(11,46,134,157,111,24,78,94)(12,148,79,57,112,85,135,35)(13,48,136,159,113,26,80,96)(14,150,61,59,114,87,137,37)(15,50,138,141,115,28,62,98)(16,152,63,41,116,89,139,39)(17,52,140,143,117,30,64,100)(18,154,65,43,118,91,121,21)(19,54,122,145,119,32,66,82)(20,156,67,45,120,93,123,23), (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), (1,98,111,151)(2,160,112,87)(3,96,113,149)(4,158,114,85)(5,94,115,147)(6,156,116,83)(7,92,117,145)(8,154,118,81)(9,90,119,143)(10,152,120,99)(11,88,101,141)(12,150,102,97)(13,86,103,159)(14,148,104,95)(15,84,105,157)(16,146,106,93)(17,82,107,155)(18,144,108,91)(19,100,109,153)(20,142,110,89)(21,121,53,75)(22,64,54,130)(23,139,55,73)(24,62,56,128)(25,137,57,71)(26,80,58,126)(27,135,59,69)(28,78,60,124)(29,133,41,67)(30,76,42,122)(31,131,43,65)(32,74,44,140)(33,129,45,63)(34,72,46,138)(35,127,47,61)(36,70,48,136)(37,125,49,79)(38,68,50,134)(39,123,51,77)(40,66,52,132)>;
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), (1,56,124,147,101,34,68,84)(2,158,69,47,102,95,125,25)(3,58,126,149,103,36,70,86)(4,160,71,49,104,97,127,27)(5,60,128,151,105,38,72,88)(6,142,73,51,106,99,129,29)(7,42,130,153,107,40,74,90)(8,144,75,53,108,81,131,31)(9,44,132,155,109,22,76,92)(10,146,77,55,110,83,133,33)(11,46,134,157,111,24,78,94)(12,148,79,57,112,85,135,35)(13,48,136,159,113,26,80,96)(14,150,61,59,114,87,137,37)(15,50,138,141,115,28,62,98)(16,152,63,41,116,89,139,39)(17,52,140,143,117,30,64,100)(18,154,65,43,118,91,121,21)(19,54,122,145,119,32,66,82)(20,156,67,45,120,93,123,23), (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), (1,98,111,151)(2,160,112,87)(3,96,113,149)(4,158,114,85)(5,94,115,147)(6,156,116,83)(7,92,117,145)(8,154,118,81)(9,90,119,143)(10,152,120,99)(11,88,101,141)(12,150,102,97)(13,86,103,159)(14,148,104,95)(15,84,105,157)(16,146,106,93)(17,82,107,155)(18,144,108,91)(19,100,109,153)(20,142,110,89)(21,121,53,75)(22,64,54,130)(23,139,55,73)(24,62,56,128)(25,137,57,71)(26,80,58,126)(27,135,59,69)(28,78,60,124)(29,133,41,67)(30,76,42,122)(31,131,43,65)(32,74,44,140)(33,129,45,63)(34,72,46,138)(35,127,47,61)(36,70,48,136)(37,125,49,79)(38,68,50,134)(39,123,51,77)(40,66,52,132) );
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)], [(1,56,124,147,101,34,68,84),(2,158,69,47,102,95,125,25),(3,58,126,149,103,36,70,86),(4,160,71,49,104,97,127,27),(5,60,128,151,105,38,72,88),(6,142,73,51,106,99,129,29),(7,42,130,153,107,40,74,90),(8,144,75,53,108,81,131,31),(9,44,132,155,109,22,76,92),(10,146,77,55,110,83,133,33),(11,46,134,157,111,24,78,94),(12,148,79,57,112,85,135,35),(13,48,136,159,113,26,80,96),(14,150,61,59,114,87,137,37),(15,50,138,141,115,28,62,98),(16,152,63,41,116,89,139,39),(17,52,140,143,117,30,64,100),(18,154,65,43,118,91,121,21),(19,54,122,145,119,32,66,82),(20,156,67,45,120,93,123,23)], [(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)], [(1,98,111,151),(2,160,112,87),(3,96,113,149),(4,158,114,85),(5,94,115,147),(6,156,116,83),(7,92,117,145),(8,154,118,81),(9,90,119,143),(10,152,120,99),(11,88,101,141),(12,150,102,97),(13,86,103,159),(14,148,104,95),(15,84,105,157),(16,146,106,93),(17,82,107,155),(18,144,108,91),(19,100,109,153),(20,142,110,89),(21,121,53,75),(22,64,54,130),(23,139,55,73),(24,62,56,128),(25,137,57,71),(26,80,58,126),(27,135,59,69),(28,78,60,124),(29,133,41,67),(30,76,42,122),(31,131,43,65),(32,74,44,140),(33,129,45,63),(34,72,46,138),(35,127,47,61),(36,70,48,136),(37,125,49,79),(38,68,50,134),(39,123,51,77),(40,66,52,132)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 40 | 2 | 2 | 8 | 8 | 10 | 10 | 10 | 10 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | D4⋊2D5 | D4×D5 | SD16⋊3D5 | Q8.D10 |
| kernel | Q8⋊Dic5⋊C2 | D20⋊6C4 | C8×Dic5 | D20⋊5C4 | Q8⋊Dic5 | C5×Q8⋊C4 | C4.Dic10 | C20.23D4 | C2×Dic5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C10 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of Q8⋊Dic5⋊C2 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 26 | 15 |
| 0 | 0 | 26 | 26 |
| 25 | 16 | 0 | 0 |
| 25 | 39 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 12 | 29 |
| 28 | 28 | 0 | 0 |
| 32 | 13 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 26 | 26 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,26,26,0,0,15,26],[25,25,0,0,16,39,0,0,0,0,12,12,0,0,12,29],[28,32,0,0,28,13,0,0,0,0,15,26,0,0,26,26] >;
Q8⋊Dic5⋊C2 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm Dic}_5\rtimes C_2 % in TeX
G:=Group("Q8:Dic5:C2"); // GroupNames label
G:=SmallGroup(320,427);
// by ID
G=gap.SmallGroup(320,427);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,120,1094,135,184,570,297,136,12550]);
// 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=d*b*d^-1=a*b^3,d*c*d^-1=a*b^4*c^9>;
// generators/relations