metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4)⋊6D20, (C2×C20)⋊28D4, (C2×C42)⋊6D5, (C2×D20)⋊18C4, C10.48(C4×D4), C2.19(C4×D20), C20⋊7(C22⋊C4), C4⋊2(D10⋊C4), C2.2(C20⋊4D4), C2.2(C20⋊7D4), (C22×D20).4C2, C22.39(C2×D20), C10.10(C4⋊1D4), C10.56(C4⋊D4), C2.3(C4.D20), (C22×C4).399D10, C10.12(C4.4D4), C22.48(C4○D20), C5⋊2(C24.3C22), (C23×D5).11C22, C23.272(C22×D5), (C22×C20).478C22, (C22×C10).314C23, (C22×Dic5).32C22, (C2×C4×C20)⋊7C2, (C2×C4⋊Dic5)⋊7C2, (C2×C4).111(C4×D5), (C2×D10⋊C4)⋊2C2, C22.119(C2×C4×D5), (C2×C20).398(C2×C4), (C2×C10).428(C2×D4), C2.6(C2×D10⋊C4), C10.74(C2×C22⋊C4), C22.43(C2×C5⋊D4), (C2×C10).73(C4○D4), (C2×C4).239(C5⋊D4), (C22×D5).23(C2×C4), (C2×C10).201(C22×C4), SmallGroup(320,566)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊6D20
G = < a,b,c,d | a2=b4=c20=d2=1, dbd=ab=ba, ac=ca, ad=da, bc=cb, dcd=c-1 >
Subgroups: 1182 in 258 conjugacy classes, 87 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C5, C2×C4 [×10], C2×C4 [×10], D4 [×8], C23, C23 [×16], D5 [×4], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C24 [×2], Dic5 [×2], C20 [×4], C20 [×4], D10 [×20], C2×C10 [×3], C2×C10 [×4], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, D20 [×8], C2×Dic5 [×6], C2×C20 [×10], C2×C20 [×4], C22×D5 [×4], C22×D5 [×12], C22×C10, C24.3C22, C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20 [×2], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5 [×2], C2×C4⋊Dic5, C2×D10⋊C4 [×4], C2×C4×C20, C22×D20, (C2×C4)⋊6D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C4⋊1D4, C4×D5 [×2], D20 [×6], C5⋊D4 [×2], C22×D5, C24.3C22, D10⋊C4 [×4], C2×C4×D5, C2×D20 [×3], C4○D20 [×2], C2×C5⋊D4, C4×D20 [×2], C20⋊4D4, C4.D20, C2×D10⋊C4, C20⋊7D4 [×2], (C2×C4)⋊6D20
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 21)(20 22)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 132)(82 133)(83 134)(84 135)(85 136)(86 137)(87 138)(88 139)(89 140)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 57 143 94)(2 58 144 95)(3 59 145 96)(4 60 146 97)(5 41 147 98)(6 42 148 99)(7 43 149 100)(8 44 150 81)(9 45 151 82)(10 46 152 83)(11 47 153 84)(12 48 154 85)(13 49 155 86)(14 50 156 87)(15 51 157 88)(16 52 158 89)(17 53 159 90)(18 54 160 91)(19 55 141 92)(20 56 142 93)(21 75 101 123)(22 76 102 124)(23 77 103 125)(24 78 104 126)(25 79 105 127)(26 80 106 128)(27 61 107 129)(28 62 108 130)(29 63 109 131)(30 64 110 132)(31 65 111 133)(32 66 112 134)(33 67 113 135)(34 68 114 136)(35 69 115 137)(36 70 116 138)(37 71 117 139)(38 72 118 140)(39 73 119 121)(40 74 120 122)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(81 91)(82 90)(83 89)(84 88)(85 87)(92 100)(93 99)(94 98)(95 97)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 143)(108 142)(109 141)(110 160)(111 159)(112 158)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(134 140)(135 139)(136 138)
G:=sub<Sym(160)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,21)(20,22)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,132)(82,133)(83,134)(84,135)(85,136)(86,137)(87,138)(88,139)(89,140)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,57,143,94)(2,58,144,95)(3,59,145,96)(4,60,146,97)(5,41,147,98)(6,42,148,99)(7,43,149,100)(8,44,150,81)(9,45,151,82)(10,46,152,83)(11,47,153,84)(12,48,154,85)(13,49,155,86)(14,50,156,87)(15,51,157,88)(16,52,158,89)(17,53,159,90)(18,54,160,91)(19,55,141,92)(20,56,142,93)(21,75,101,123)(22,76,102,124)(23,77,103,125)(24,78,104,126)(25,79,105,127)(26,80,106,128)(27,61,107,129)(28,62,108,130)(29,63,109,131)(30,64,110,132)(31,65,111,133)(32,66,112,134)(33,67,113,135)(34,68,114,136)(35,69,115,137)(36,70,116,138)(37,71,117,139)(38,72,118,140)(39,73,119,121)(40,74,120,122), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138)>;
G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,21)(20,22)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(81,132)(82,133)(83,134)(84,135)(85,136)(86,137)(87,138)(88,139)(89,140)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,141)(102,142)(103,143)(104,144)(105,145)(106,146)(107,147)(108,148)(109,149)(110,150)(111,151)(112,152)(113,153)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160), (1,57,143,94)(2,58,144,95)(3,59,145,96)(4,60,146,97)(5,41,147,98)(6,42,148,99)(7,43,149,100)(8,44,150,81)(9,45,151,82)(10,46,152,83)(11,47,153,84)(12,48,154,85)(13,49,155,86)(14,50,156,87)(15,51,157,88)(16,52,158,89)(17,53,159,90)(18,54,160,91)(19,55,141,92)(20,56,142,93)(21,75,101,123)(22,76,102,124)(23,77,103,125)(24,78,104,126)(25,79,105,127)(26,80,106,128)(27,61,107,129)(28,62,108,130)(29,63,109,131)(30,64,110,132)(31,65,111,133)(32,66,112,134)(33,67,113,135)(34,68,114,136)(35,69,115,137)(36,70,116,138)(37,71,117,139)(38,72,118,140)(39,73,119,121)(40,74,120,122), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(41,57)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(58,60)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,160)(111,159)(112,158)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(134,140)(135,139)(136,138) );
G=PermutationGroup([(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,21),(20,22),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(81,132),(82,133),(83,134),(84,135),(85,136),(86,137),(87,138),(88,139),(89,140),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(97,128),(98,129),(99,130),(100,131),(101,141),(102,142),(103,143),(104,144),(105,145),(106,146),(107,147),(108,148),(109,149),(110,150),(111,151),(112,152),(113,153),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160)], [(1,57,143,94),(2,58,144,95),(3,59,145,96),(4,60,146,97),(5,41,147,98),(6,42,148,99),(7,43,149,100),(8,44,150,81),(9,45,151,82),(10,46,152,83),(11,47,153,84),(12,48,154,85),(13,49,155,86),(14,50,156,87),(15,51,157,88),(16,52,158,89),(17,53,159,90),(18,54,160,91),(19,55,141,92),(20,56,142,93),(21,75,101,123),(22,76,102,124),(23,77,103,125),(24,78,104,126),(25,79,105,127),(26,80,106,128),(27,61,107,129),(28,62,108,130),(29,63,109,131),(30,64,110,132),(31,65,111,133),(32,66,112,134),(33,67,113,135),(34,68,114,136),(35,69,115,137),(36,70,116,138),(37,71,117,139),(38,72,118,140),(39,73,119,121),(40,74,120,122)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(41,57),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(58,60),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(78,80),(81,91),(82,90),(83,89),(84,88),(85,87),(92,100),(93,99),(94,98),(95,97),(101,149),(102,148),(103,147),(104,146),(105,145),(106,144),(107,143),(108,142),(109,141),(110,160),(111,159),(112,158),(113,157),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(134,140),(135,139),(136,138)])
92 conjugacy classes
class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
type | + | + | + | + | + | + | + | + | + | |||||
image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D20 | C5⋊D4 | C4○D20 |
kernel | (C2×C4)⋊6D20 | C2×C4⋊Dic5 | C2×D10⋊C4 | C2×C4×C20 | C22×D20 | C2×D20 | C2×C20 | C2×C42 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
# reps | 1 | 1 | 4 | 1 | 1 | 8 | 8 | 2 | 4 | 6 | 8 | 24 | 8 | 16 |
Matrix representation of (C2×C4)⋊6D20 ►in GL6(𝔽41)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 40 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 9 | 0 | 0 |
0 | 0 | 0 | 0 | 30 | 32 |
0 | 0 | 0 | 0 | 9 | 11 |
16 | 30 | 0 | 0 | 0 | 0 |
27 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 35 | 1 | 0 | 0 |
0 | 0 | 5 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 9 | 11 |
0 | 0 | 0 | 0 | 30 | 14 |
40 | 40 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 40 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 40 |
0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[16,27,0,0,0,0,30,2,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
(C2×C4)⋊6D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_6D_{20}
% in TeX
G:=Group("(C2xC4):6D20");
// GroupNames label
G:=SmallGroup(320,566);
// by ID
G=gap.SmallGroup(320,566);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,d*b*d=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*c*d=c^-1>;
// generators/relations