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G = C2×C20⋊D4  order 320 = 26·5

Direct product of C2 and C20⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C20⋊D4, C24.40D10, C20⋊9(C2×D4), (C2×C20)⋊13D4, (C2×D4)⋊39D10, Dic5⋊2(C2×D4), (C22×D4)⋊9D5, C10⋊2(C4⋊1D4), (C2×Dic5)⋊14D4, (C22×D20)⋊19C2, (C2×D20)⋊56C22, (D4×C10)⋊44C22, C22.149(D4×D5), (C2×C10).298C24, (C2×C20).544C23, (C4×Dic5)⋊68C22, C10.145(C22×D4), (C22×C4).380D10, (C23×C10).78C22, (C23×D5).77C22, C23.135(C22×D5), C22.311(C23×D5), (C22×C10).232C23, (C22×C20).276C22, (C2×Dic5).295C23, (C22×D5).129C23, (C22×Dic5).255C22, (D4×C2×C10)⋊6C2, C4⋊1(C2×C5⋊D4), C5⋊3(C2×C4⋊1D4), C2.105(C2×D4×D5), (C2×C4×Dic5)⋊12C2, (C2×C4)⋊10(C5⋊D4), (C2×C10).581(C2×D4), (C2×C5⋊D4)⋊47C22, (C22×C5⋊D4)⋊16C2, C2.18(C22×C5⋊D4), (C2×C4).627(C22×D5), C22.111(C2×C5⋊D4), SmallGroup(320,1475)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×C20⋊D4
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×D20 — C2×C20⋊D4
Lower central
C5 — C2×C10 — C2×C20⋊D4

Subgroups: 2014 in 498 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×40], C5, C2×C4 [×6], C2×C4 [×12], D4 [×48], C23, C23 [×4], C23 [×28], D5 [×4], C10, C10 [×6], C10 [×4], C42 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×44], C24 [×2], C24 [×2], Dic5 [×8], C20 [×4], D10 [×20], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C42, C4⋊1D4 [×8], C22×D4, C22×D4 [×5], D20 [×8], C2×Dic5 [×12], C5⋊D4 [×32], C2×C20 [×6], C5×D4 [×8], C22×D5 [×4], C22×D5 [×12], C22×C10, C22×C10 [×4], C22×C10 [×12], C2×C4⋊1D4, C4×Dic5 [×4], C2×D20 [×4], C2×D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×16], C2×C5⋊D4 [×16], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5 [×2], C23×C10 [×2], C2×C4×Dic5, C20⋊D4 [×8], C22×D20, C22×C5⋊D4 [×4], D4×C2×C10, C2×C20⋊D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D5, C2×D4 [×18], C24, D10 [×7], C4⋊1D4 [×4], C22×D4 [×3], C5⋊D4 [×4], C22×D5 [×7], C2×C4⋊1D4, D4×D5 [×4], C2×C5⋊D4 [×6], C23×D5, C20⋊D4 [×4], C2×D4×D5 [×2], C22×C5⋊D4, C2×C20⋊D4

Generators and relations
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=c-1 >

Smallest permutation representation
►On 160 points
Generators in S160
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(81 160)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)

(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)

(1 103 93 59)(2 112 94 48)(3 101 95 57)(4 110 96 46)(5 119 97 55)(6 108 98 44)(7 117 99 53)(8 106 100 42)(9 115 81 51)(10 104 82 60)(11 113 83 49)(12 102 84 58)(13 111 85 47)(14 120 86 56)(15 109 87 45)(16 118 88 54)(17 107 89 43)(18 116 90 52)(19 105 91 41)(20 114 92 50)(21 73 137 158)(22 62 138 147)(23 71 139 156)(24 80 140 145)(25 69 121 154)(26 78 122 143)(27 67 123 152)(28 76 124 141)(29 65 125 150)(30 74 126 159)(31 63 127 148)(32 72 128 157)(33 61 129 146)(34 70 130 155)(35 79 131 144)(36 68 132 153)(37 77 133 142)(38 66 134 151)(39 75 135 160)(40 64 136 149)

(1 83)(2 82)(3 81)(4 100)(5 99)(6 98)(7 97)(8 96)(9 95)(10 94)(11 93)(12 92)(13 91)(14 90)(15 89)(16 88)(17 87)(18 86)(19 85)(20 84)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 148)(62 147)(63 146)(64 145)(65 144)(66 143)(67 142)(68 141)(69 160)(70 159)(71 158)(72 157)(73 156)(74 155)(75 154)(76 153)(77 152)(78 151)(79 150)(80 149)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(116 120)(117 119)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)
 
G:=sub<Sym(160)| (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140), (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), (1,103,93,59)(2,112,94,48)(3,101,95,57)(4,110,96,46)(5,119,97,55)(6,108,98,44)(7,117,99,53)(8,106,100,42)(9,115,81,51)(10,104,82,60)(11,113,83,49)(12,102,84,58)(13,111,85,47)(14,120,86,56)(15,109,87,45)(16,118,88,54)(17,107,89,43)(18,116,90,52)(19,105,91,41)(20,114,92,50)(21,73,137,158)(22,62,138,147)(23,71,139,156)(24,80,140,145)(25,69,121,154)(26,78,122,143)(27,67,123,152)(28,76,124,141)(29,65,125,150)(30,74,126,159)(31,63,127,148)(32,72,128,157)(33,61,129,146)(34,70,130,155)(35,79,131,144)(36,68,132,153)(37,77,133,142)(38,66,134,151)(39,75,135,160)(40,64,136,149), (1,83)(2,82)(3,81)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,84)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,160)(70,159)(71,158)(72,157)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(116,120)(117,119)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)>;
 
G:=Group( (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140), (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), (1,103,93,59)(2,112,94,48)(3,101,95,57)(4,110,96,46)(5,119,97,55)(6,108,98,44)(7,117,99,53)(8,106,100,42)(9,115,81,51)(10,104,82,60)(11,113,83,49)(12,102,84,58)(13,111,85,47)(14,120,86,56)(15,109,87,45)(16,118,88,54)(17,107,89,43)(18,116,90,52)(19,105,91,41)(20,114,92,50)(21,73,137,158)(22,62,138,147)(23,71,139,156)(24,80,140,145)(25,69,121,154)(26,78,122,143)(27,67,123,152)(28,76,124,141)(29,65,125,150)(30,74,126,159)(31,63,127,148)(32,72,128,157)(33,61,129,146)(34,70,130,155)(35,79,131,144)(36,68,132,153)(37,77,133,142)(38,66,134,151)(39,75,135,160)(40,64,136,149), (1,83)(2,82)(3,81)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,84)(21,23)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,160)(70,159)(71,158)(72,157)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(116,120)(117,119)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139) );
 
G=PermutationGroup([(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(81,160),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140)], [(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)], [(1,103,93,59),(2,112,94,48),(3,101,95,57),(4,110,96,46),(5,119,97,55),(6,108,98,44),(7,117,99,53),(8,106,100,42),(9,115,81,51),(10,104,82,60),(11,113,83,49),(12,102,84,58),(13,111,85,47),(14,120,86,56),(15,109,87,45),(16,118,88,54),(17,107,89,43),(18,116,90,52),(19,105,91,41),(20,114,92,50),(21,73,137,158),(22,62,138,147),(23,71,139,156),(24,80,140,145),(25,69,121,154),(26,78,122,143),(27,67,123,152),(28,76,124,141),(29,65,125,150),(30,74,126,159),(31,63,127,148),(32,72,128,157),(33,61,129,146),(34,70,130,155),(35,79,131,144),(36,68,132,153),(37,77,133,142),(38,66,134,151),(39,75,135,160),(40,64,136,149)], [(1,83),(2,82),(3,81),(4,100),(5,99),(6,98),(7,97),(8,96),(9,95),(10,94),(11,93),(12,92),(13,91),(14,90),(15,89),(16,88),(17,87),(18,86),(19,85),(20,84),(21,23),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,148),(62,147),(63,146),(64,145),(65,144),(66,143),(67,142),(68,141),(69,160),(70,159),(71,158),(72,157),(73,156),(74,155),(75,154),(76,153),(77,152),(78,151),(79,150),(80,149),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109),(116,120),(117,119),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,140),(137,139)])
 

Matrix representation ►G ⊆ GL6(𝔽41)

100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
0040100
0053500
0000402
0000401
,
25320000
24160000
00211700
00152000
0000139
0000140
,
4000000
4010000
006100
0063500
000010
0000140

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[25,24,0,0,0,0,32,16,0,0,0,0,0,0,21,15,0,0,0,0,17,20,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4L5A5B10A···10N10O···10AD20A···20H
order12···22222222244444···45510···1010···1020···20
size11···1444420202020222210···10222···24···44···4

68 irreducible representations

dim11111122222224
type+++++++++++++
imageC1C2C2C2C2C2D4D4D5D10D10D10C5⋊D4D4×D5
kernelC2×C20⋊D4C2×C4×Dic5C20⋊D4C22×D20C22×C5⋊D4D4×C2×C10C2×Dic5C2×C20C22×D4C22×C4C2×D4C24C2×C4C22
# reps118141842284168

In GAP, Magma, Sage, TeX 

C_2\times C_{20}\rtimes D_4
 
% in TeX
 
G:=Group("C2xC20:D4");
 
// GroupNames label
 
G:=SmallGroup(320,1475);
 
// by ID
 
G=gap.SmallGroup(320,1475);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,297,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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×
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