direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D8⋊C4, D8⋊3C20, C8⋊3(C2×C20), C40⋊31(C2×C4), (C5×D8)⋊15C4, D4⋊2(C2×C20), (C4×D4)⋊2C10, C8⋊C4⋊2C10, C4.Q8⋊3C10, (D4×C20)⋊31C2, (C2×D8).6C10, C2.17(D4×C20), (C10×D8).13C2, (C2×C20).458D4, C10.149(C4×D4), D4⋊C4⋊16C10, C42.10(C2×C10), C4.14(C22×C20), C22.56(D4×C10), C20.261(C4○D4), (C4×C20).251C22, (C2×C40).331C22, C20.218(C22×C4), (C2×C20).909C23, C10.130(C8⋊C22), (D4×C10).293C22, C4.6(C5×C4○D4), (C5×D4)⋊25(C2×C4), (C5×C8⋊C4)⋊11C2, (C5×C4.Q8)⋊12C2, C2.5(C5×C8⋊C22), C4⋊C4.50(C2×C10), (C2×C8).20(C2×C10), (C2×C4).104(C5×D4), (C5×D4⋊C4)⋊39C2, (C2×D4).51(C2×C10), (C2×C10).632(C2×D4), (C5×C4⋊C4).371C22, (C2×C4).84(C22×C10), SmallGroup(320,943)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C2 — C22 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×D4⋊C4 — C5×D8⋊C4 |
Generators and relations for C5×D8⋊C4
G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >
Subgroups: 250 in 132 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], D4 [×2], C23 [×2], C10, C10 [×2], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×8], C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, C40 [×2], C40, C2×C20, C2×C20 [×2], C2×C20 [×6], C5×D4 [×4], C5×D4 [×2], C22×C10 [×2], D8⋊C4, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], C5×D8 [×4], C22×C20 [×2], D4×C10 [×2], C5×C8⋊C4, C5×D4⋊C4 [×2], C5×C4.Q8, D4×C20 [×2], C10×D8, C5×D8⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C8⋊C22 [×2], C2×C20 [×6], C5×D4 [×2], C22×C10, D8⋊C4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8⋊C22 [×2], C5×D8⋊C4
(1 127 159 36 151)(2 128 160 37 152)(3 121 153 38 145)(4 122 154 39 146)(5 123 155 40 147)(6 124 156 33 148)(7 125 157 34 149)(8 126 158 35 150)(9 116 140 17 132)(10 117 141 18 133)(11 118 142 19 134)(12 119 143 20 135)(13 120 144 21 136)(14 113 137 22 129)(15 114 138 23 130)(16 115 139 24 131)(25 85 54 48 93)(26 86 55 41 94)(27 87 56 42 95)(28 88 49 43 96)(29 81 50 44 89)(30 82 51 45 90)(31 83 52 46 91)(32 84 53 47 92)(57 75 107 67 102)(58 76 108 68 103)(59 77 109 69 104)(60 78 110 70 97)(61 79 111 71 98)(62 80 112 72 99)(63 73 105 65 100)(64 74 106 66 101)
(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 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 22)(18 21)(19 20)(23 24)(25 26)(27 32)(28 31)(29 30)(33 38)(34 37)(35 36)(39 40)(41 48)(42 47)(43 46)(44 45)(49 52)(50 51)(53 56)(54 55)(57 60)(58 59)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 82)(83 88)(84 87)(85 86)(89 90)(91 96)(92 95)(93 94)(97 102)(98 101)(99 100)(103 104)(105 112)(106 111)(107 110)(108 109)(113 116)(114 115)(117 120)(118 119)(121 124)(122 123)(125 128)(126 127)(129 132)(130 131)(133 136)(134 135)(137 140)(138 139)(141 144)(142 143)(145 148)(146 147)(149 152)(150 151)(153 156)(154 155)(157 160)(158 159)
(1 119 41 69)(2 116 42 66)(3 113 43 71)(4 118 44 68)(5 115 45 65)(6 120 46 70)(7 117 47 67)(8 114 48 72)(9 56 106 152)(10 53 107 149)(11 50 108 146)(12 55 109 151)(13 52 110 148)(14 49 111 145)(15 54 112 150)(16 51 105 147)(17 27 64 160)(18 32 57 157)(19 29 58 154)(20 26 59 159)(21 31 60 156)(22 28 61 153)(23 25 62 158)(24 30 63 155)(33 136 83 78)(34 133 84 75)(35 130 85 80)(36 135 86 77)(37 132 87 74)(38 129 88 79)(39 134 81 76)(40 131 82 73)(89 103 122 142)(90 100 123 139)(91 97 124 144)(92 102 125 141)(93 99 126 138)(94 104 127 143)(95 101 128 140)(96 98 121 137)
G:=sub<Sym(160)| (1,127,159,36,151)(2,128,160,37,152)(3,121,153,38,145)(4,122,154,39,146)(5,123,155,40,147)(6,124,156,33,148)(7,125,157,34,149)(8,126,158,35,150)(9,116,140,17,132)(10,117,141,18,133)(11,118,142,19,134)(12,119,143,20,135)(13,120,144,21,136)(14,113,137,22,129)(15,114,138,23,130)(16,115,139,24,131)(25,85,54,48,93)(26,86,55,41,94)(27,87,56,42,95)(28,88,49,43,96)(29,81,50,44,89)(30,82,51,45,90)(31,83,52,46,91)(32,84,53,47,92)(57,75,107,67,102)(58,76,108,68,103)(59,77,109,69,104)(60,78,110,70,97)(61,79,111,71,98)(62,80,112,72,99)(63,73,105,65,100)(64,74,106,66,101), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,22)(18,21)(19,20)(23,24)(25,26)(27,32)(28,31)(29,30)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,88)(84,87)(85,86)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,112)(106,111)(107,110)(108,109)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159), (1,119,41,69)(2,116,42,66)(3,113,43,71)(4,118,44,68)(5,115,45,65)(6,120,46,70)(7,117,47,67)(8,114,48,72)(9,56,106,152)(10,53,107,149)(11,50,108,146)(12,55,109,151)(13,52,110,148)(14,49,111,145)(15,54,112,150)(16,51,105,147)(17,27,64,160)(18,32,57,157)(19,29,58,154)(20,26,59,159)(21,31,60,156)(22,28,61,153)(23,25,62,158)(24,30,63,155)(33,136,83,78)(34,133,84,75)(35,130,85,80)(36,135,86,77)(37,132,87,74)(38,129,88,79)(39,134,81,76)(40,131,82,73)(89,103,122,142)(90,100,123,139)(91,97,124,144)(92,102,125,141)(93,99,126,138)(94,104,127,143)(95,101,128,140)(96,98,121,137)>;
G:=Group( (1,127,159,36,151)(2,128,160,37,152)(3,121,153,38,145)(4,122,154,39,146)(5,123,155,40,147)(6,124,156,33,148)(7,125,157,34,149)(8,126,158,35,150)(9,116,140,17,132)(10,117,141,18,133)(11,118,142,19,134)(12,119,143,20,135)(13,120,144,21,136)(14,113,137,22,129)(15,114,138,23,130)(16,115,139,24,131)(25,85,54,48,93)(26,86,55,41,94)(27,87,56,42,95)(28,88,49,43,96)(29,81,50,44,89)(30,82,51,45,90)(31,83,52,46,91)(32,84,53,47,92)(57,75,107,67,102)(58,76,108,68,103)(59,77,109,69,104)(60,78,110,70,97)(61,79,111,71,98)(62,80,112,72,99)(63,73,105,65,100)(64,74,106,66,101), (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), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,16)(17,22)(18,21)(19,20)(23,24)(25,26)(27,32)(28,31)(29,30)(33,38)(34,37)(35,36)(39,40)(41,48)(42,47)(43,46)(44,45)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(73,80)(74,79)(75,78)(76,77)(81,82)(83,88)(84,87)(85,86)(89,90)(91,96)(92,95)(93,94)(97,102)(98,101)(99,100)(103,104)(105,112)(106,111)(107,110)(108,109)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159), (1,119,41,69)(2,116,42,66)(3,113,43,71)(4,118,44,68)(5,115,45,65)(6,120,46,70)(7,117,47,67)(8,114,48,72)(9,56,106,152)(10,53,107,149)(11,50,108,146)(12,55,109,151)(13,52,110,148)(14,49,111,145)(15,54,112,150)(16,51,105,147)(17,27,64,160)(18,32,57,157)(19,29,58,154)(20,26,59,159)(21,31,60,156)(22,28,61,153)(23,25,62,158)(24,30,63,155)(33,136,83,78)(34,133,84,75)(35,130,85,80)(36,135,86,77)(37,132,87,74)(38,129,88,79)(39,134,81,76)(40,131,82,73)(89,103,122,142)(90,100,123,139)(91,97,124,144)(92,102,125,141)(93,99,126,138)(94,104,127,143)(95,101,128,140)(96,98,121,137) );
G=PermutationGroup([(1,127,159,36,151),(2,128,160,37,152),(3,121,153,38,145),(4,122,154,39,146),(5,123,155,40,147),(6,124,156,33,148),(7,125,157,34,149),(8,126,158,35,150),(9,116,140,17,132),(10,117,141,18,133),(11,118,142,19,134),(12,119,143,20,135),(13,120,144,21,136),(14,113,137,22,129),(15,114,138,23,130),(16,115,139,24,131),(25,85,54,48,93),(26,86,55,41,94),(27,87,56,42,95),(28,88,49,43,96),(29,81,50,44,89),(30,82,51,45,90),(31,83,52,46,91),(32,84,53,47,92),(57,75,107,67,102),(58,76,108,68,103),(59,77,109,69,104),(60,78,110,70,97),(61,79,111,71,98),(62,80,112,72,99),(63,73,105,65,100),(64,74,106,66,101)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,16),(17,22),(18,21),(19,20),(23,24),(25,26),(27,32),(28,31),(29,30),(33,38),(34,37),(35,36),(39,40),(41,48),(42,47),(43,46),(44,45),(49,52),(50,51),(53,56),(54,55),(57,60),(58,59),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69),(73,80),(74,79),(75,78),(76,77),(81,82),(83,88),(84,87),(85,86),(89,90),(91,96),(92,95),(93,94),(97,102),(98,101),(99,100),(103,104),(105,112),(106,111),(107,110),(108,109),(113,116),(114,115),(117,120),(118,119),(121,124),(122,123),(125,128),(126,127),(129,132),(130,131),(133,136),(134,135),(137,140),(138,139),(141,144),(142,143),(145,148),(146,147),(149,152),(150,151),(153,156),(154,155),(157,160),(158,159)], [(1,119,41,69),(2,116,42,66),(3,113,43,71),(4,118,44,68),(5,115,45,65),(6,120,46,70),(7,117,47,67),(8,114,48,72),(9,56,106,152),(10,53,107,149),(11,50,108,146),(12,55,109,151),(13,52,110,148),(14,49,111,145),(15,54,112,150),(16,51,105,147),(17,27,64,160),(18,32,57,157),(19,29,58,154),(20,26,59,159),(21,31,60,156),(22,28,61,153),(23,25,62,158),(24,30,63,155),(33,136,83,78),(34,133,84,75),(35,130,85,80),(36,135,86,77),(37,132,87,74),(38,129,88,79),(39,134,81,76),(40,131,82,73),(89,103,122,142),(90,100,123,139),(91,97,124,144),(92,102,125,141),(93,99,126,138),(94,104,127,143),(95,101,128,140),(96,98,121,137)])
110 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20X | 20Y | ··· | 20AN | 40A | ··· | 40P |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | ||||||||||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | C8⋊C22 | C5×C8⋊C22 |
kernel | C5×D8⋊C4 | C5×C8⋊C4 | C5×D4⋊C4 | C5×C4.Q8 | D4×C20 | C10×D8 | C5×D8 | D8⋊C4 | C8⋊C4 | D4⋊C4 | C4.Q8 | C4×D4 | C2×D8 | D8 | C2×C20 | C20 | C2×C4 | C4 | C10 | C2 |
# reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 8 | 4 | 32 | 2 | 2 | 8 | 8 | 2 | 8 |
Matrix representation of C5×D8⋊C4 ►in GL6(𝔽41)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 37 | 0 | 0 | 0 |
0 | 0 | 0 | 37 | 0 | 0 |
0 | 0 | 0 | 0 | 37 | 0 |
0 | 0 | 0 | 0 | 0 | 37 |
0 | 32 | 0 | 0 | 0 | 0 |
32 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 35 | 12 | 0 | 31 |
0 | 0 | 21 | 29 | 31 | 10 |
0 | 0 | 36 | 14 | 0 | 26 |
0 | 0 | 37 | 40 | 29 | 18 |
0 | 32 | 0 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 35 | 12 | 0 | 31 |
0 | 0 | 20 | 12 | 10 | 31 |
0 | 0 | 36 | 14 | 0 | 26 |
0 | 0 | 7 | 40 | 12 | 35 |
9 | 0 | 0 | 0 | 0 | 0 |
0 | 9 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 26 | 8 | 1 | 39 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 2 | 11 | 17 | 33 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,35,21,36,37,0,0,12,29,14,40,0,0,0,31,0,29,0,0,31,10,26,18],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,35,20,36,7,0,0,12,12,14,40,0,0,0,10,0,12,0,0,31,31,26,35],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,26,1,2,0,0,0,8,0,11,0,0,1,1,0,17,0,0,0,39,0,33] >;
C5×D8⋊C4 in GAP, Magma, Sage, TeX
C_5\times D_8\rtimes C_4
% in TeX
G:=Group("C5xD8:C4");
// GroupNames label
G:=SmallGroup(320,943);
// by ID
G=gap.SmallGroup(320,943);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,1276,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations