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## G = D4⋊6D20order 320 = 26·5

### 2nd semidirect product of D4 and D20 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D4⋊6D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×D4⋊2D5 — D4⋊6D20
 Lower central C5 — C2×C10 — D4⋊6D20
 Upper central C1 — C22 — C4×D4

Generators and relations for D46D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 1030 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic5 [×6], C20 [×4], C20 [×3], D10 [×6], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×8], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×D5 [×2], C22×C10 [×2], D46D4, C4⋊Dic5, C4⋊Dic5 [×8], D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, D42D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C202Q8, C4×D20, C22.D20 [×4], D102Q8 [×2], C2×C4⋊Dic5 [×2], C207D4 [×2], D4×C20, C2×D42D5 [×2], D46D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, D20 [×4], C22×D5 [×7], D46D4, C2×D20 [×6], D42D5 [×2], C23×D5, C22×D20, C2×D42D5, D4.10D10, D46D20

Smallest permutation representation of D46D20
On 160 points
Generators in S160
(1 159 69 24)(2 160 70 25)(3 141 71 26)(4 142 72 27)(5 143 73 28)(6 144 74 29)(7 145 75 30)(8 146 76 31)(9 147 77 32)(10 148 78 33)(11 149 79 34)(12 150 80 35)(13 151 61 36)(14 152 62 37)(15 153 63 38)(16 154 64 39)(17 155 65 40)(18 156 66 21)(19 157 67 22)(20 158 68 23)(41 109 136 99)(42 110 137 100)(43 111 138 81)(44 112 139 82)(45 113 140 83)(46 114 121 84)(47 115 122 85)(48 116 123 86)(49 117 124 87)(50 118 125 88)(51 119 126 89)(52 120 127 90)(53 101 128 91)(54 102 129 92)(55 103 130 93)(56 104 131 94)(57 105 132 95)(58 106 133 96)(59 107 134 97)(60 108 135 98)
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 121)(19 122)(20 123)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 81)(39 82)(40 83)(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)(101 143)(102 144)(103 145)(104 146)(105 147)(106 148)(107 149)(108 150)(109 151)(110 152)(111 153)(112 154)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 141)(120 142)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 123)(50 122)(51 121)(52 140)(53 139)(54 138)(55 137)(56 136)(57 135)(58 134)(59 133)(60 132)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 102)(82 101)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 160)(158 159)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 20 20 2 2 2 2 4 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 D10 D20 2- 1+4 D4⋊2D5 D4.10D10 kernel D4⋊6D20 C20⋊2Q8 C4×D20 C22.D20 D10⋊2Q8 C2×C4⋊Dic5 C20⋊7D4 D4×C20 C2×D4⋊2D5 C5×D4 C4×D4 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C10 C4 C2 # reps 1 1 1 4 2 2 2 1 2 4 2 4 2 4 2 4 2 16 1 4 4

Matrix representation of D46D20 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 32 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 0 9 0 0 32 0
,
 27 2 0 0 25 11 0 0 0 0 40 0 0 0 0 40
,
 30 2 0 0 22 11 0 0 0 0 40 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,9,0],[27,25,0,0,2,11,0,0,0,0,40,0,0,0,0,40],[30,22,0,0,2,11,0,0,0,0,40,0,0,0,0,1] >;

D46D20 in GAP, Magma, Sage, TeX

D_4\rtimes_6D_{20}
% in TeX

G:=Group("D4:6D20");
// GroupNames label

G:=SmallGroup(320,1227);
// by ID

G=gap.SmallGroup(320,1227);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,675,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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