direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D5⋊C8, C42.17F5, C20.26C42, D10.8C42, C20⋊4(C2×C8), D5⋊1(C4×C8), (C4×D5)⋊6C8, C4.19(C4×F5), (C4×C20).16C4, Dic5⋊6(C2×C8), D10.12(C2×C8), C10.1(C2×C42), C10.1(C22×C8), (C4×Dic5).43C4, (D5×C42).28C2, C22.24(C22×F5), Dic5.25(C22×C4), (C2×Dic5).311C23, (C4×Dic5).353C22, C5⋊1(C2×C4×C8), C5⋊C8⋊7(C2×C4), (C4×C5⋊C8)⋊19C2, C2.1(C2×C4×F5), C2.1(C2×D5⋊C8), (C2×C4×D5).42C4, (C2×D5⋊C8).11C2, (C4×D5).69(C2×C4), (C2×C5⋊C8).43C22, (C2×C4).159(C2×F5), (C2×C20).165(C2×C4), (C2×C4×D5).408C22, (C2×C10).13(C22×C4), (C2×Dic5).161(C2×C4), (C22×D5).113(C2×C4), SmallGroup(320,1013)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C4×D5⋊C8 |
Generators and relations for C4×D5⋊C8
G = < a,b,c,d | a4=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >
Subgroups: 426 in 162 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C4×C8, C2×C42, C22×C8, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C2×C4×C8, C4×Dic5, C4×Dic5, C4×C20, D5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C4×C5⋊C8, D5×C42, C2×D5⋊C8, C4×D5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, F5, C4×C8, C2×C42, C22×C8, C2×F5, C2×C4×C8, D5⋊C8, C4×F5, C22×F5, C2×D5⋊C8, C2×C4×F5, C4×D5⋊C8
►On 160 points
Generators in S160
(1 12 60 75)(2 13 61 76)(3 14 62 77)(4 15 63 78)(5 16 64 79)(6 9 57 80)(7 10 58 73)(8 11 59 74)(17 68 83 33)(18 69 84 34)(19 70 85 35)(20 71 86 36)(21 72 87 37)(22 65 88 38)(23 66 81 39)(24 67 82 40)(25 114 106 90)(26 115 107 91)(27 116 108 92)(28 117 109 93)(29 118 110 94)(30 119 111 95)(31 120 112 96)(32 113 105 89)(41 142 126 104)(42 143 127 97)(43 144 128 98)(44 137 121 99)(45 138 122 100)(46 139 123 101)(47 140 124 102)(48 141 125 103)(49 151 157 133)(50 152 158 134)(51 145 159 135)(52 146 160 136)(53 147 153 129)(54 148 154 130)(55 149 155 131)(56 150 156 132)
(1 66 109 144 135)(2 137 67 136 110)(3 129 138 111 68)(4 112 130 69 139)(5 70 105 140 131)(6 141 71 132 106)(7 133 142 107 72)(8 108 134 65 143)(9 125 86 56 90)(10 49 126 91 87)(11 92 50 88 127)(12 81 93 128 51)(13 121 82 52 94)(14 53 122 95 83)(15 96 54 84 123)(16 85 89 124 55)(17 77 153 45 119)(18 46 78 120 154)(19 113 47 155 79)(20 156 114 80 48)(21 73 157 41 115)(22 42 74 116 158)(23 117 43 159 75)(24 160 118 76 44)(25 57 103 36 150)(26 37 58 151 104)(27 152 38 97 59)(28 98 145 60 39)(29 61 99 40 146)(30 33 62 147 100)(31 148 34 101 63)(32 102 149 64 35)
(1 135)(2 110)(3 68)(4 139)(5 131)(6 106)(7 72)(8 143)(9 90)(10 87)(11 127)(12 51)(13 94)(14 83)(15 123)(16 55)(17 77)(18 120)(19 47)(21 73)(22 116)(23 43)(25 57)(26 151)(27 38)(29 61)(30 147)(31 34)(33 62)(35 102)(37 58)(39 98)(42 74)(44 160)(46 78)(48 156)(49 91)(52 121)(53 95)(56 125)(59 97)(60 145)(63 101)(64 149)(65 108)(66 144)(69 112)(70 140)(75 159)(76 118)(79 155)(80 114)(81 128)(84 96)(85 124)(88 92)(99 146)(103 150)(107 133)(111 129)(115 157)(119 153)(132 141)(136 137)
(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)| (1,12,60,75)(2,13,61,76)(3,14,62,77)(4,15,63,78)(5,16,64,79)(6,9,57,80)(7,10,58,73)(8,11,59,74)(17,68,83,33)(18,69,84,34)(19,70,85,35)(20,71,86,36)(21,72,87,37)(22,65,88,38)(23,66,81,39)(24,67,82,40)(25,114,106,90)(26,115,107,91)(27,116,108,92)(28,117,109,93)(29,118,110,94)(30,119,111,95)(31,120,112,96)(32,113,105,89)(41,142,126,104)(42,143,127,97)(43,144,128,98)(44,137,121,99)(45,138,122,100)(46,139,123,101)(47,140,124,102)(48,141,125,103)(49,151,157,133)(50,152,158,134)(51,145,159,135)(52,146,160,136)(53,147,153,129)(54,148,154,130)(55,149,155,131)(56,150,156,132), (1,66,109,144,135)(2,137,67,136,110)(3,129,138,111,68)(4,112,130,69,139)(5,70,105,140,131)(6,141,71,132,106)(7,133,142,107,72)(8,108,134,65,143)(9,125,86,56,90)(10,49,126,91,87)(11,92,50,88,127)(12,81,93,128,51)(13,121,82,52,94)(14,53,122,95,83)(15,96,54,84,123)(16,85,89,124,55)(17,77,153,45,119)(18,46,78,120,154)(19,113,47,155,79)(20,156,114,80,48)(21,73,157,41,115)(22,42,74,116,158)(23,117,43,159,75)(24,160,118,76,44)(25,57,103,36,150)(26,37,58,151,104)(27,152,38,97,59)(28,98,145,60,39)(29,61,99,40,146)(30,33,62,147,100)(31,148,34,101,63)(32,102,149,64,35), (1,135)(2,110)(3,68)(4,139)(5,131)(6,106)(7,72)(8,143)(9,90)(10,87)(11,127)(12,51)(13,94)(14,83)(15,123)(16,55)(17,77)(18,120)(19,47)(21,73)(22,116)(23,43)(25,57)(26,151)(27,38)(29,61)(30,147)(31,34)(33,62)(35,102)(37,58)(39,98)(42,74)(44,160)(46,78)(48,156)(49,91)(52,121)(53,95)(56,125)(59,97)(60,145)(63,101)(64,149)(65,108)(66,144)(69,112)(70,140)(75,159)(76,118)(79,155)(80,114)(81,128)(84,96)(85,124)(88,92)(99,146)(103,150)(107,133)(111,129)(115,157)(119,153)(132,141)(136,137), (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( (1,12,60,75)(2,13,61,76)(3,14,62,77)(4,15,63,78)(5,16,64,79)(6,9,57,80)(7,10,58,73)(8,11,59,74)(17,68,83,33)(18,69,84,34)(19,70,85,35)(20,71,86,36)(21,72,87,37)(22,65,88,38)(23,66,81,39)(24,67,82,40)(25,114,106,90)(26,115,107,91)(27,116,108,92)(28,117,109,93)(29,118,110,94)(30,119,111,95)(31,120,112,96)(32,113,105,89)(41,142,126,104)(42,143,127,97)(43,144,128,98)(44,137,121,99)(45,138,122,100)(46,139,123,101)(47,140,124,102)(48,141,125,103)(49,151,157,133)(50,152,158,134)(51,145,159,135)(52,146,160,136)(53,147,153,129)(54,148,154,130)(55,149,155,131)(56,150,156,132), (1,66,109,144,135)(2,137,67,136,110)(3,129,138,111,68)(4,112,130,69,139)(5,70,105,140,131)(6,141,71,132,106)(7,133,142,107,72)(8,108,134,65,143)(9,125,86,56,90)(10,49,126,91,87)(11,92,50,88,127)(12,81,93,128,51)(13,121,82,52,94)(14,53,122,95,83)(15,96,54,84,123)(16,85,89,124,55)(17,77,153,45,119)(18,46,78,120,154)(19,113,47,155,79)(20,156,114,80,48)(21,73,157,41,115)(22,42,74,116,158)(23,117,43,159,75)(24,160,118,76,44)(25,57,103,36,150)(26,37,58,151,104)(27,152,38,97,59)(28,98,145,60,39)(29,61,99,40,146)(30,33,62,147,100)(31,148,34,101,63)(32,102,149,64,35), (1,135)(2,110)(3,68)(4,139)(5,131)(6,106)(7,72)(8,143)(9,90)(10,87)(11,127)(12,51)(13,94)(14,83)(15,123)(16,55)(17,77)(18,120)(19,47)(21,73)(22,116)(23,43)(25,57)(26,151)(27,38)(29,61)(30,147)(31,34)(33,62)(35,102)(37,58)(39,98)(42,74)(44,160)(46,78)(48,156)(49,91)(52,121)(53,95)(56,125)(59,97)(60,145)(63,101)(64,149)(65,108)(66,144)(69,112)(70,140)(75,159)(76,118)(79,155)(80,114)(81,128)(84,96)(85,124)(88,92)(99,146)(103,150)(107,133)(111,129)(115,157)(119,153)(132,141)(136,137), (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([[(1,12,60,75),(2,13,61,76),(3,14,62,77),(4,15,63,78),(5,16,64,79),(6,9,57,80),(7,10,58,73),(8,11,59,74),(17,68,83,33),(18,69,84,34),(19,70,85,35),(20,71,86,36),(21,72,87,37),(22,65,88,38),(23,66,81,39),(24,67,82,40),(25,114,106,90),(26,115,107,91),(27,116,108,92),(28,117,109,93),(29,118,110,94),(30,119,111,95),(31,120,112,96),(32,113,105,89),(41,142,126,104),(42,143,127,97),(43,144,128,98),(44,137,121,99),(45,138,122,100),(46,139,123,101),(47,140,124,102),(48,141,125,103),(49,151,157,133),(50,152,158,134),(51,145,159,135),(52,146,160,136),(53,147,153,129),(54,148,154,130),(55,149,155,131),(56,150,156,132)], [(1,66,109,144,135),(2,137,67,136,110),(3,129,138,111,68),(4,112,130,69,139),(5,70,105,140,131),(6,141,71,132,106),(7,133,142,107,72),(8,108,134,65,143),(9,125,86,56,90),(10,49,126,91,87),(11,92,50,88,127),(12,81,93,128,51),(13,121,82,52,94),(14,53,122,95,83),(15,96,54,84,123),(16,85,89,124,55),(17,77,153,45,119),(18,46,78,120,154),(19,113,47,155,79),(20,156,114,80,48),(21,73,157,41,115),(22,42,74,116,158),(23,117,43,159,75),(24,160,118,76,44),(25,57,103,36,150),(26,37,58,151,104),(27,152,38,97,59),(28,98,145,60,39),(29,61,99,40,146),(30,33,62,147,100),(31,148,34,101,63),(32,102,149,64,35)], [(1,135),(2,110),(3,68),(4,139),(5,131),(6,106),(7,72),(8,143),(9,90),(10,87),(11,127),(12,51),(13,94),(14,83),(15,123),(16,55),(17,77),(18,120),(19,47),(21,73),(22,116),(23,43),(25,57),(26,151),(27,38),(29,61),(30,147),(31,34),(33,62),(35,102),(37,58),(39,98),(42,74),(44,160),(46,78),(48,156),(49,91),(52,121),(53,95),(56,125),(59,97),(60,145),(63,101),(64,149),(65,108),(66,144),(69,112),(70,140),(75,159),(76,118),(79,155),(80,114),(81,128),(84,96),(85,124),(88,92),(99,146),(103,150),(107,133),(111,129),(115,157),(119,153),(132,141),(136,137)], [(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 | ··· | 4L | 4M | ··· | 4X | 5 | 8A | ··· | 8AF | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | F5 | C2×F5 | D5⋊C8 | C4×F5 |
| kernel | C4×D5⋊C8 | C4×C5⋊C8 | D5×C42 | C2×D5⋊C8 | C4×Dic5 | C4×C20 | D5⋊C8 | C2×C4×D5 | C4×D5 | C42 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 2 | 16 | 4 | 32 | 1 | 3 | 8 | 4 |
Matrix representation of C4×D5⋊C8 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 34 | 7 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 34 | 7 | 0 | 0 |
| 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 7 | 34 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 32 | 22 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,40,34,0,0,0,0,0,40,34,0,0,0,7,7],[40,0,0,0,0,0,34,40,0,0,0,7,7,0,0,0,0,0,7,7,0,0,0,40,34],[27,0,0,0,0,0,0,0,32,0,0,0,0,22,9,0,9,0,0,0,0,0,9,0,0] >;
C4×D5⋊C8 in GAP, Magma, Sage, TeX
C_4\times D_5\rtimes C_8
% in TeX
G:=Group("C4xD5:C8"); // GroupNames label
G:=SmallGroup(320,1013);
// by ID
G=gap.SmallGroup(320,1013);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^5=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations