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G = (C2×C4)⋊6D20  order 320 = 26·5

1st semidirect product of C2×C4 and D20 acting via D20/C20=C2

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

Derived series
C1 — C2×C10 — (C2×C4)⋊6D20
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C22×D20 — (C2×C4)⋊6D20
Lower central
C5 — C2×C10 — (C2×C4)⋊6D20
Upper central
C1 — C23 — C2×C42

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

Smallest permutation representation of (C2×C4)⋊6D20
►On 160 points
Generators in S160
(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···2G2H2I2J2K4A···4L4M4N4O4P5A5B10A···10N20A···20AV
order12···222224···444445510···1020···20
size11···1202020202···220202020222···22···2

92 irreducible representations

dim11111122222222
type+++++++++
imageC1C2C2C2C2C4D4D5C4○D4D10C4×D5D20C5⋊D4C4○D20
kernel(C2×C4)⋊6D20C2×C4⋊Dic5C2×D10⋊C4C2×C4×C20C22×D20C2×D20C2×C20C2×C42C2×C10C22×C4C2×C4C2×C4C2×C4C22
# reps1141188246824816

Matrix representation of (C2×C4)⋊6D20 ►in GL6(𝔽41)

100000
010000
001000
000100
0000400
0000040
,
100000
010000
009000
000900
00003032
0000911
,
16300000
2720000
0035100
0054000
0000911
00003014
,
40400000
010000
00404000
000100
0000040
0000400

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
 

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×
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ℤ
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