direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.47D4, M4(2).32D10, C4.67(C2×D20), (C2×C4).53D20, (C2×C20).176D4, C20.422(C2×D4), C23.56(C4×D5), C10⋊2(C4.10D4), (C2×C20).418C23, (C2×Dic10).28C4, (C22×C4).144D10, (C2×M4(2)).17D5, C4.29(D10⋊C4), (C22×Dic5).5C4, C20.101(C22⋊C4), (C10×M4(2)).28C2, C4.Dic5.43C22, (C22×C20).191C22, (C22×Dic10).16C2, (C5×M4(2)).35C22, C22.51(D10⋊C4), (C2×Dic10).287C22, (C2×C4).55(C4×D5), C5⋊4(C2×C4.10D4), C22.22(C2×C4×D5), C4.115(C2×C5⋊D4), (C2×C20).284(C2×C4), (C2×Dic5).6(C2×C4), C2.33(C2×D10⋊C4), (C2×C4).257(C5⋊D4), C10.102(C2×C22⋊C4), (C2×C4).122(C22×D5), (C2×C4.Dic5).26C2, (C22×C10).140(C2×C4), (C2×C10).117(C22×C4), (C2×C10).131(C22⋊C4), SmallGroup(320,763)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20.47D4
G = < a,b,c,d | a2=b20=1, c4=d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c3 >
Subgroups: 478 in 146 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×6], C2×C4 [×8], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×8], Dic5 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×2], C4.10D4 [×4], C2×M4(2), C2×M4(2), C22×Q8, C5⋊2C8 [×2], C40 [×2], Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C2×C4.10D4, C2×C5⋊2C8, C4.Dic5 [×2], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C20.47D4 [×4], C2×C4.Dic5, C10×M4(2), C22×Dic10, C2×C20.47D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4.10D4 [×2], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C4.10D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C20.47D4 [×2], C2×D10⋊C4, C2×C20.47D4
(1 40)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 101)(60 102)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 141)(78 142)(79 143)(80 144)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)
(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 44 25 101 11 54 35 111)(2 43 26 120 12 53 36 110)(3 42 27 119 13 52 37 109)(4 41 28 118 14 51 38 108)(5 60 29 117 15 50 39 107)(6 59 30 116 16 49 40 106)(7 58 31 115 17 48 21 105)(8 57 32 114 18 47 22 104)(9 56 33 113 19 46 23 103)(10 55 34 112 20 45 24 102)(61 121 160 98 71 131 150 88)(62 140 141 97 72 130 151 87)(63 139 142 96 73 129 152 86)(64 138 143 95 74 128 153 85)(65 137 144 94 75 127 154 84)(66 136 145 93 76 126 155 83)(67 135 146 92 77 125 156 82)(68 134 147 91 78 124 157 81)(69 133 148 90 79 123 158 100)(70 132 149 89 80 122 159 99)
(1 129 11 139)(2 128 12 138)(3 127 13 137)(4 126 14 136)(5 125 15 135)(6 124 16 134)(7 123 17 133)(8 122 18 132)(9 121 19 131)(10 140 20 130)(21 100 31 90)(22 99 32 89)(23 98 33 88)(24 97 34 87)(25 96 35 86)(26 95 36 85)(27 94 37 84)(28 93 38 83)(29 92 39 82)(30 91 40 81)(41 61 51 71)(42 80 52 70)(43 79 53 69)(44 78 54 68)(45 77 55 67)(46 76 56 66)(47 75 57 65)(48 74 58 64)(49 73 59 63)(50 72 60 62)(101 147 111 157)(102 146 112 156)(103 145 113 155)(104 144 114 154)(105 143 115 153)(106 142 116 152)(107 141 117 151)(108 160 118 150)(109 159 119 149)(110 158 120 148)
G:=sub<Sym(160)| (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,141)(78,142)(79,143)(80,144)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128), (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), (1,44,25,101,11,54,35,111)(2,43,26,120,12,53,36,110)(3,42,27,119,13,52,37,109)(4,41,28,118,14,51,38,108)(5,60,29,117,15,50,39,107)(6,59,30,116,16,49,40,106)(7,58,31,115,17,48,21,105)(8,57,32,114,18,47,22,104)(9,56,33,113,19,46,23,103)(10,55,34,112,20,45,24,102)(61,121,160,98,71,131,150,88)(62,140,141,97,72,130,151,87)(63,139,142,96,73,129,152,86)(64,138,143,95,74,128,153,85)(65,137,144,94,75,127,154,84)(66,136,145,93,76,126,155,83)(67,135,146,92,77,125,156,82)(68,134,147,91,78,124,157,81)(69,133,148,90,79,123,158,100)(70,132,149,89,80,122,159,99), (1,129,11,139)(2,128,12,138)(3,127,13,137)(4,126,14,136)(5,125,15,135)(6,124,16,134)(7,123,17,133)(8,122,18,132)(9,121,19,131)(10,140,20,130)(21,100,31,90)(22,99,32,89)(23,98,33,88)(24,97,34,87)(25,96,35,86)(26,95,36,85)(27,94,37,84)(28,93,38,83)(29,92,39,82)(30,91,40,81)(41,61,51,71)(42,80,52,70)(43,79,53,69)(44,78,54,68)(45,77,55,67)(46,76,56,66)(47,75,57,65)(48,74,58,64)(49,73,59,63)(50,72,60,62)(101,147,111,157)(102,146,112,156)(103,145,113,155)(104,144,114,154)(105,143,115,153)(106,142,116,152)(107,141,117,151)(108,160,118,150)(109,159,119,149)(110,158,120,148)>;
G:=Group( (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,101)(60,102)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,141)(78,142)(79,143)(80,144)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128), (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), (1,44,25,101,11,54,35,111)(2,43,26,120,12,53,36,110)(3,42,27,119,13,52,37,109)(4,41,28,118,14,51,38,108)(5,60,29,117,15,50,39,107)(6,59,30,116,16,49,40,106)(7,58,31,115,17,48,21,105)(8,57,32,114,18,47,22,104)(9,56,33,113,19,46,23,103)(10,55,34,112,20,45,24,102)(61,121,160,98,71,131,150,88)(62,140,141,97,72,130,151,87)(63,139,142,96,73,129,152,86)(64,138,143,95,74,128,153,85)(65,137,144,94,75,127,154,84)(66,136,145,93,76,126,155,83)(67,135,146,92,77,125,156,82)(68,134,147,91,78,124,157,81)(69,133,148,90,79,123,158,100)(70,132,149,89,80,122,159,99), (1,129,11,139)(2,128,12,138)(3,127,13,137)(4,126,14,136)(5,125,15,135)(6,124,16,134)(7,123,17,133)(8,122,18,132)(9,121,19,131)(10,140,20,130)(21,100,31,90)(22,99,32,89)(23,98,33,88)(24,97,34,87)(25,96,35,86)(26,95,36,85)(27,94,37,84)(28,93,38,83)(29,92,39,82)(30,91,40,81)(41,61,51,71)(42,80,52,70)(43,79,53,69)(44,78,54,68)(45,77,55,67)(46,76,56,66)(47,75,57,65)(48,74,58,64)(49,73,59,63)(50,72,60,62)(101,147,111,157)(102,146,112,156)(103,145,113,155)(104,144,114,154)(105,143,115,153)(106,142,116,152)(107,141,117,151)(108,160,118,150)(109,159,119,149)(110,158,120,148) );
G=PermutationGroup([(1,40),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,101),(60,102),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(67,151),(68,152),(69,153),(70,154),(71,155),(72,156),(73,157),(74,158),(75,159),(76,160),(77,141),(78,142),(79,143),(80,144),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128)], [(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)], [(1,44,25,101,11,54,35,111),(2,43,26,120,12,53,36,110),(3,42,27,119,13,52,37,109),(4,41,28,118,14,51,38,108),(5,60,29,117,15,50,39,107),(6,59,30,116,16,49,40,106),(7,58,31,115,17,48,21,105),(8,57,32,114,18,47,22,104),(9,56,33,113,19,46,23,103),(10,55,34,112,20,45,24,102),(61,121,160,98,71,131,150,88),(62,140,141,97,72,130,151,87),(63,139,142,96,73,129,152,86),(64,138,143,95,74,128,153,85),(65,137,144,94,75,127,154,84),(66,136,145,93,76,126,155,83),(67,135,146,92,77,125,156,82),(68,134,147,91,78,124,157,81),(69,133,148,90,79,123,158,100),(70,132,149,89,80,122,159,99)], [(1,129,11,139),(2,128,12,138),(3,127,13,137),(4,126,14,136),(5,125,15,135),(6,124,16,134),(7,123,17,133),(8,122,18,132),(9,121,19,131),(10,140,20,130),(21,100,31,90),(22,99,32,89),(23,98,33,88),(24,97,34,87),(25,96,35,86),(26,95,36,85),(27,94,37,84),(28,93,38,83),(29,92,39,82),(30,91,40,81),(41,61,51,71),(42,80,52,70),(43,79,53,69),(44,78,54,68),(45,77,55,67),(46,76,56,66),(47,75,57,65),(48,74,58,64),(49,73,59,63),(50,72,60,62),(101,147,111,157),(102,146,112,156),(103,145,113,155),(104,144,114,154),(105,143,115,153),(106,142,116,152),(107,141,117,151),(108,160,118,150),(109,159,119,149),(110,158,120,148)])
62 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | - | - | |||||
image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C4×D5 | C4.10D4 | C20.47D4 |
kernel | C2×C20.47D4 | C20.47D4 | C2×C4.Dic5 | C10×M4(2) | C22×Dic10 | C2×Dic10 | C22×Dic5 | C2×C20 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C2 |
# reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 2 | 8 |
Matrix representation of C2×C20.47D4 ►in GL6(𝔽41)
40 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
1 | 40 | 0 | 0 | 0 | 0 |
36 | 6 | 0 | 0 | 0 | 0 |
0 | 0 | 9 | 11 | 0 | 0 |
0 | 0 | 30 | 14 | 0 | 0 |
0 | 0 | 0 | 0 | 39 | 11 |
0 | 0 | 0 | 0 | 14 | 25 |
35 | 3 | 0 | 0 | 0 | 0 |
15 | 6 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 24 | 4 | 39 |
0 | 0 | 17 | 40 | 27 | 39 |
0 | 0 | 8 | 17 | 23 | 1 |
0 | 0 | 10 | 9 | 3 | 18 |
28 | 27 | 0 | 0 | 0 | 0 |
12 | 13 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 32 | 0 | 0 |
0 | 0 | 32 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 32 | 0 |
0 | 0 | 0 | 0 | 4 | 9 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[35,15,0,0,0,0,3,6,0,0,0,0,0,0,1,17,8,10,0,0,24,40,17,9,0,0,4,27,23,3,0,0,39,39,1,18],[28,12,0,0,0,0,27,13,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,32,4,0,0,0,0,0,9] >;
C2×C20.47D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}._{47}D_4
% in TeX
G:=Group("C2xC20.47D4");
// GroupNames label
G:=SmallGroup(320,763);
// by ID
G=gap.SmallGroup(320,763);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,58,1123,136,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=1,c^4=d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^3>;
// generators/relations