direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D5⋊C8, C42.17F5, C20.26C42, D10.8C42, C20⋊4(C2×C8), (C4×D5)⋊6C8, D5⋊1(C4×C8), C4.19(C4×F5), (C4×C20).16C4, Dic5⋊6(C2×C8), D10.12(C2×C8), C10.1(C22×C8), C10.1(C2×C42), (C4×Dic5).43C4, (D5×C42).28C2, C22.24(C22×F5), Dic5.25(C22×C4), (C4×Dic5).353C22, (C2×Dic5).311C23, 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 |
Subgroups: 426 in 162 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C4 [×6], C22, C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×2], C2×C4 [×15], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C2×C8 [×12], C22×C4 [×3], Dic5 [×6], C20 [×6], D10 [×6], C2×C10, C4×C8 [×4], C2×C42, C22×C8 [×2], C5⋊C8 [×8], C4×D5 [×12], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C2×C4×C8, C4×Dic5, C4×Dic5 [×2], C4×C20, D5⋊C8 [×8], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], C4×C5⋊C8 [×4], D5×C42, C2×D5⋊C8 [×2], C4×D5⋊C8
Quotients:
C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], F5, C4×C8 [×4], C2×C42, C22×C8 [×2], C2×F5 [×3], C2×C4×C8, D5⋊C8 [×4], C4×F5 [×2], C22×F5, C2×D5⋊C8 [×2], C2×C4×F5, C4×D5⋊C8
Generators and relations
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 >
►On 160 points
Generators in S160
(1 12 105 75)(2 13 106 76)(3 14 107 77)(4 15 108 78)(5 16 109 79)(6 9 110 80)(7 10 111 73)(8 11 112 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 58 90)(26 115 59 91)(27 116 60 92)(28 117 61 93)(29 118 62 94)(30 119 63 95)(31 120 64 96)(32 113 57 89)(41 129 126 104)(42 130 127 97)(43 131 128 98)(44 132 121 99)(45 133 122 100)(46 134 123 101)(47 135 124 102)(48 136 125 103)(49 151 157 144)(50 152 158 137)(51 145 159 138)(52 146 160 139)(53 147 153 140)(54 148 154 141)(55 149 155 142)(56 150 156 143)
(1 66 61 131 138)(2 132 67 139 62)(3 140 133 63 68)(4 64 141 69 134)(5 70 57 135 142)(6 136 71 143 58)(7 144 129 59 72)(8 60 137 65 130)(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)(25 110 103 36 150)(26 37 111 151 104)(27 152 38 97 112)(28 98 145 105 39)(29 106 99 40 146)(30 33 107 147 100)(31 148 34 101 108)(32 102 149 109 35)
(1 138)(2 62)(3 68)(4 134)(5 142)(6 58)(7 72)(8 130)(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)(25 110)(26 151)(27 38)(29 106)(30 147)(31 34)(33 107)(35 102)(37 111)(39 98)(42 74)(44 160)(46 78)(48 156)(49 91)(52 121)(53 95)(56 125)(59 144)(60 65)(63 140)(64 69)(66 131)(70 135)(75 159)(76 118)(79 155)(80 114)(81 128)(84 96)(85 124)(88 92)(97 112)(99 146)(101 108)(103 150)(105 145)(109 149)(115 157)(119 153)(132 139)(136 143)
(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)| (1,12,105,75)(2,13,106,76)(3,14,107,77)(4,15,108,78)(5,16,109,79)(6,9,110,80)(7,10,111,73)(8,11,112,74)(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)(25,114,58,90)(26,115,59,91)(27,116,60,92)(28,117,61,93)(29,118,62,94)(30,119,63,95)(31,120,64,96)(32,113,57,89)(41,129,126,104)(42,130,127,97)(43,131,128,98)(44,132,121,99)(45,133,122,100)(46,134,123,101)(47,135,124,102)(48,136,125,103)(49,151,157,144)(50,152,158,137)(51,145,159,138)(52,146,160,139)(53,147,153,140)(54,148,154,141)(55,149,155,142)(56,150,156,143), (1,66,61,131,138)(2,132,67,139,62)(3,140,133,63,68)(4,64,141,69,134)(5,70,57,135,142)(6,136,71,143,58)(7,144,129,59,72)(8,60,137,65,130)(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)(25,110,103,36,150)(26,37,111,151,104)(27,152,38,97,112)(28,98,145,105,39)(29,106,99,40,146)(30,33,107,147,100)(31,148,34,101,108)(32,102,149,109,35), (1,138)(2,62)(3,68)(4,134)(5,142)(6,58)(7,72)(8,130)(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)(25,110)(26,151)(27,38)(29,106)(30,147)(31,34)(33,107)(35,102)(37,111)(39,98)(42,74)(44,160)(46,78)(48,156)(49,91)(52,121)(53,95)(56,125)(59,144)(60,65)(63,140)(64,69)(66,131)(70,135)(75,159)(76,118)(79,155)(80,114)(81,128)(84,96)(85,124)(88,92)(97,112)(99,146)(101,108)(103,150)(105,145)(109,149)(115,157)(119,153)(132,139)(136,143), (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( (1,12,105,75)(2,13,106,76)(3,14,107,77)(4,15,108,78)(5,16,109,79)(6,9,110,80)(7,10,111,73)(8,11,112,74)(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)(25,114,58,90)(26,115,59,91)(27,116,60,92)(28,117,61,93)(29,118,62,94)(30,119,63,95)(31,120,64,96)(32,113,57,89)(41,129,126,104)(42,130,127,97)(43,131,128,98)(44,132,121,99)(45,133,122,100)(46,134,123,101)(47,135,124,102)(48,136,125,103)(49,151,157,144)(50,152,158,137)(51,145,159,138)(52,146,160,139)(53,147,153,140)(54,148,154,141)(55,149,155,142)(56,150,156,143), (1,66,61,131,138)(2,132,67,139,62)(3,140,133,63,68)(4,64,141,69,134)(5,70,57,135,142)(6,136,71,143,58)(7,144,129,59,72)(8,60,137,65,130)(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)(25,110,103,36,150)(26,37,111,151,104)(27,152,38,97,112)(28,98,145,105,39)(29,106,99,40,146)(30,33,107,147,100)(31,148,34,101,108)(32,102,149,109,35), (1,138)(2,62)(3,68)(4,134)(5,142)(6,58)(7,72)(8,130)(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)(25,110)(26,151)(27,38)(29,106)(30,147)(31,34)(33,107)(35,102)(37,111)(39,98)(42,74)(44,160)(46,78)(48,156)(49,91)(52,121)(53,95)(56,125)(59,144)(60,65)(63,140)(64,69)(66,131)(70,135)(75,159)(76,118)(79,155)(80,114)(81,128)(84,96)(85,124)(88,92)(97,112)(99,146)(101,108)(103,150)(105,145)(109,149)(115,157)(119,153)(132,139)(136,143), (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([(1,12,105,75),(2,13,106,76),(3,14,107,77),(4,15,108,78),(5,16,109,79),(6,9,110,80),(7,10,111,73),(8,11,112,74),(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),(25,114,58,90),(26,115,59,91),(27,116,60,92),(28,117,61,93),(29,118,62,94),(30,119,63,95),(31,120,64,96),(32,113,57,89),(41,129,126,104),(42,130,127,97),(43,131,128,98),(44,132,121,99),(45,133,122,100),(46,134,123,101),(47,135,124,102),(48,136,125,103),(49,151,157,144),(50,152,158,137),(51,145,159,138),(52,146,160,139),(53,147,153,140),(54,148,154,141),(55,149,155,142),(56,150,156,143)], [(1,66,61,131,138),(2,132,67,139,62),(3,140,133,63,68),(4,64,141,69,134),(5,70,57,135,142),(6,136,71,143,58),(7,144,129,59,72),(8,60,137,65,130),(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),(25,110,103,36,150),(26,37,111,151,104),(27,152,38,97,112),(28,98,145,105,39),(29,106,99,40,146),(30,33,107,147,100),(31,148,34,101,108),(32,102,149,109,35)], [(1,138),(2,62),(3,68),(4,134),(5,142),(6,58),(7,72),(8,130),(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),(25,110),(26,151),(27,38),(29,106),(30,147),(31,34),(33,107),(35,102),(37,111),(39,98),(42,74),(44,160),(46,78),(48,156),(49,91),(52,121),(53,95),(56,125),(59,144),(60,65),(63,140),(64,69),(66,131),(70,135),(75,159),(76,118),(79,155),(80,114),(81,128),(84,96),(85,124),(88,92),(97,112),(99,146),(101,108),(103,150),(105,145),(109,149),(115,157),(119,153),(132,139),(136,143)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀