direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×C5⋊2C8, C42.205D10, C5⋊7(C8×D4), C20⋊7(C2×C8), (C5×D4)⋊4C8, C20⋊3C8⋊20C2, (C4×D4).15D5, (D4×C20).5C2, C4.214(D4×D5), C2.3(D4×Dic5), (D4×C10).21C4, C10.120(C4×D4), C20.373(C2×D4), C4⋊C4.11Dic5, C10.62(C8○D4), C10.48(C22×C8), (C4×C20).81C22, (C2×D4).11Dic5, C22⋊C4.7Dic5, C20.306(C4○D4), C20.55D4⋊24C2, (C2×C20).848C23, (C22×C4).309D10, C2.2(D4.Dic5), C4.133(D4⋊2D5), C23.16(C2×Dic5), (C22×C20).348C22, C22.22(C22×Dic5), C4⋊1(C2×C5⋊2C8), (C4×C5⋊2C8)⋊6C2, (C2×C10)⋊7(C2×C8), (C5×C4⋊C4).23C4, C22⋊1(C2×C5⋊2C8), C2.5(C22×C5⋊2C8), (C2×C20).335(C2×C4), (C22×C5⋊2C8)⋊18C2, (C5×C22⋊C4).16C4, (C2×C4).33(C2×Dic5), (C2×C4).790(C22×D5), (C22×C10).127(C2×C4), (C2×C10).286(C22×C4), (C2×C5⋊2C8).323C22, SmallGroup(320,637)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C5⋊2C8
G = < a,b,c,d | a4=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 286 in 134 conjugacy classes, 77 normal (33 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C8×D4, C2×C5⋊2C8, C2×C5⋊2C8, C2×C5⋊2C8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C4×C5⋊2C8, C20⋊3C8, C20.55D4, C22×C5⋊2C8, D4×C20, D4×C5⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D5, C2×C8, C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C22×C8, C8○D4, C5⋊2C8, C2×Dic5, C22×D5, C8×D4, C2×C5⋊2C8, D4×D5, D4⋊2D5, C22×Dic5, C22×C5⋊2C8, D4×Dic5, D4.Dic5, D4×C5⋊2C8
►On 160 points
Generators in S160
(1 40 137 22)(2 33 138 23)(3 34 139 24)(4 35 140 17)(5 36 141 18)(6 37 142 19)(7 38 143 20)(8 39 144 21)(9 42 134 104)(10 43 135 97)(11 44 136 98)(12 45 129 99)(13 46 130 100)(14 47 131 101)(15 48 132 102)(16 41 133 103)(25 123 151 63)(26 124 152 64)(27 125 145 57)(28 126 146 58)(29 127 147 59)(30 128 148 60)(31 121 149 61)(32 122 150 62)(49 156 106 118)(50 157 107 119)(51 158 108 120)(52 159 109 113)(53 160 110 114)(54 153 111 115)(55 154 112 116)(56 155 105 117)(65 73 83 92)(66 74 84 93)(67 75 85 94)(68 76 86 95)(69 77 87 96)(70 78 88 89)(71 79 81 90)(72 80 82 91)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 145)(32 146)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 97)(48 98)(49 110)(50 111)(51 112)(52 105)(53 106)(54 107)(55 108)(56 109)(57 61)(58 62)(59 63)(60 64)(65 87)(66 88)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 77)(74 78)(75 79)(76 80)(89 93)(90 94)(91 95)(92 96)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(153 157)(154 158)(155 159)(156 160)
(1 94 132 63 113)(2 114 64 133 95)(3 96 134 57 115)(4 116 58 135 89)(5 90 136 59 117)(6 118 60 129 91)(7 92 130 61 119)(8 120 62 131 93)(9 125 153 139 77)(10 78 140 154 126)(11 127 155 141 79)(12 80 142 156 128)(13 121 157 143 73)(14 74 144 158 122)(15 123 159 137 75)(16 76 138 160 124)(17 112 146 43 88)(18 81 44 147 105)(19 106 148 45 82)(20 83 46 149 107)(21 108 150 47 84)(22 85 48 151 109)(23 110 152 41 86)(24 87 42 145 111)(25 52 40 67 102)(26 103 68 33 53)(27 54 34 69 104)(28 97 70 35 55)(29 56 36 71 98)(30 99 72 37 49)(31 50 38 65 100)(32 101 66 39 51)
(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)
G:=sub<Sym(160)| (1,40,137,22)(2,33,138,23)(3,34,139,24)(4,35,140,17)(5,36,141,18)(6,37,142,19)(7,38,143,20)(8,39,144,21)(9,42,134,104)(10,43,135,97)(11,44,136,98)(12,45,129,99)(13,46,130,100)(14,47,131,101)(15,48,132,102)(16,41,133,103)(25,123,151,63)(26,124,152,64)(27,125,145,57)(28,126,146,58)(29,127,147,59)(30,128,148,60)(31,121,149,61)(32,122,150,62)(49,156,106,118)(50,157,107,119)(51,158,108,120)(52,159,109,113)(53,160,110,114)(54,153,111,115)(55,154,112,116)(56,155,105,117)(65,73,83,92)(66,74,84,93)(67,75,85,94)(68,76,86,95)(69,77,87,96)(70,78,88,89)(71,79,81,90)(72,80,82,91), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,145)(32,146)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,61)(58,62)(59,63)(60,64)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(153,157)(154,158)(155,159)(156,160), (1,94,132,63,113)(2,114,64,133,95)(3,96,134,57,115)(4,116,58,135,89)(5,90,136,59,117)(6,118,60,129,91)(7,92,130,61,119)(8,120,62,131,93)(9,125,153,139,77)(10,78,140,154,126)(11,127,155,141,79)(12,80,142,156,128)(13,121,157,143,73)(14,74,144,158,122)(15,123,159,137,75)(16,76,138,160,124)(17,112,146,43,88)(18,81,44,147,105)(19,106,148,45,82)(20,83,46,149,107)(21,108,150,47,84)(22,85,48,151,109)(23,110,152,41,86)(24,87,42,145,111)(25,52,40,67,102)(26,103,68,33,53)(27,54,34,69,104)(28,97,70,35,55)(29,56,36,71,98)(30,99,72,37,49)(31,50,38,65,100)(32,101,66,39,51), (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)>;
G:=Group( (1,40,137,22)(2,33,138,23)(3,34,139,24)(4,35,140,17)(5,36,141,18)(6,37,142,19)(7,38,143,20)(8,39,144,21)(9,42,134,104)(10,43,135,97)(11,44,136,98)(12,45,129,99)(13,46,130,100)(14,47,131,101)(15,48,132,102)(16,41,133,103)(25,123,151,63)(26,124,152,64)(27,125,145,57)(28,126,146,58)(29,127,147,59)(30,128,148,60)(31,121,149,61)(32,122,150,62)(49,156,106,118)(50,157,107,119)(51,158,108,120)(52,159,109,113)(53,160,110,114)(54,153,111,115)(55,154,112,116)(56,155,105,117)(65,73,83,92)(66,74,84,93)(67,75,85,94)(68,76,86,95)(69,77,87,96)(70,78,88,89)(71,79,81,90)(72,80,82,91), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,145)(32,146)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,61)(58,62)(59,63)(60,64)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(153,157)(154,158)(155,159)(156,160), (1,94,132,63,113)(2,114,64,133,95)(3,96,134,57,115)(4,116,58,135,89)(5,90,136,59,117)(6,118,60,129,91)(7,92,130,61,119)(8,120,62,131,93)(9,125,153,139,77)(10,78,140,154,126)(11,127,155,141,79)(12,80,142,156,128)(13,121,157,143,73)(14,74,144,158,122)(15,123,159,137,75)(16,76,138,160,124)(17,112,146,43,88)(18,81,44,147,105)(19,106,148,45,82)(20,83,46,149,107)(21,108,150,47,84)(22,85,48,151,109)(23,110,152,41,86)(24,87,42,145,111)(25,52,40,67,102)(26,103,68,33,53)(27,54,34,69,104)(28,97,70,35,55)(29,56,36,71,98)(30,99,72,37,49)(31,50,38,65,100)(32,101,66,39,51), (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) );
G=PermutationGroup([[(1,40,137,22),(2,33,138,23),(3,34,139,24),(4,35,140,17),(5,36,141,18),(6,37,142,19),(7,38,143,20),(8,39,144,21),(9,42,134,104),(10,43,135,97),(11,44,136,98),(12,45,129,99),(13,46,130,100),(14,47,131,101),(15,48,132,102),(16,41,133,103),(25,123,151,63),(26,124,152,64),(27,125,145,57),(28,126,146,58),(29,127,147,59),(30,128,148,60),(31,121,149,61),(32,122,150,62),(49,156,106,118),(50,157,107,119),(51,158,108,120),(52,159,109,113),(53,160,110,114),(54,153,111,115),(55,154,112,116),(56,155,105,117),(65,73,83,92),(66,74,84,93),(67,75,85,94),(68,76,86,95),(69,77,87,96),(70,78,88,89),(71,79,81,90),(72,80,82,91)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,147),(26,148),(27,149),(28,150),(29,151),(30,152),(31,145),(32,146),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,97),(48,98),(49,110),(50,111),(51,112),(52,105),(53,106),(54,107),(55,108),(56,109),(57,61),(58,62),(59,63),(60,64),(65,87),(66,88),(67,81),(68,82),(69,83),(70,84),(71,85),(72,86),(73,77),(74,78),(75,79),(76,80),(89,93),(90,94),(91,95),(92,96),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(153,157),(154,158),(155,159),(156,160)], [(1,94,132,63,113),(2,114,64,133,95),(3,96,134,57,115),(4,116,58,135,89),(5,90,136,59,117),(6,118,60,129,91),(7,92,130,61,119),(8,120,62,131,93),(9,125,153,139,77),(10,78,140,154,126),(11,127,155,141,79),(12,80,142,156,128),(13,121,157,143,73),(14,74,144,158,122),(15,123,159,137,75),(16,76,138,160,124),(17,112,146,43,88),(18,81,44,147,105),(19,106,148,45,82),(20,83,46,149,107),(21,108,150,47,84),(22,85,48,151,109),(23,110,152,41,86),(24,87,42,145,111),(25,52,40,67,102),(26,103,68,33,53),(27,54,34,69,104),(28,97,70,35,55),(29,56,36,71,98),(30,99,72,37,49),(31,50,38,65,100),(32,101,66,39,51)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8T | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D5 | C4○D4 | D10 | Dic5 | Dic5 | D10 | Dic5 | C8○D4 | C5⋊2C8 | D4×D5 | D4⋊2D5 | D4.Dic5 |
| kernel | D4×C5⋊2C8 | C4×C5⋊2C8 | C20⋊3C8 | C20.55D4 | C22×C5⋊2C8 | D4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | D4×C10 | C5×D4 | C5⋊2C8 | C4×D4 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 16 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of D4×C5⋊2C8 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 22 |
| 0 | 0 | 26 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 26 | 1 |
| 10 | 0 | 0 | 0 |
| 29 | 37 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 10 | 0 | 0 |
| 32 | 9 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 14 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,26,0,0,22,1],[40,0,0,0,0,40,0,0,0,0,40,26,0,0,0,1],[10,29,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[32,32,0,0,10,9,0,0,0,0,14,0,0,0,0,14] >;
D4×C5⋊2C8 in GAP, Magma, Sage, TeX
D_4\times C_5\rtimes_2C_8
% in TeX
G:=Group("D4xC5:2C8"); // GroupNames label
G:=SmallGroup(320,637);
// by ID
G=gap.SmallGroup(320,637);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^5=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations