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G = C4○D4×C20order 320 = 26·5

Direct product of C20 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C20
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C22⋊C4 — D4×C20 — C4○D4×C20
 Lower central C1 — C2 — C4○D4×C20
 Upper central C1 — C4×C20 — C4○D4×C20

Generators and relations for C4○D4×C20
G = < a,b,c,d | a20=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 370 in 310 conjugacy classes, 250 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×6], C5, C2×C4, C2×C4 [×23], C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×12], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×C20, C2×C20 [×23], C2×C20 [×12], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C4×C4○D4, C4×C20, C4×C20 [×9], C5×C22⋊C4 [×6], C5×C4⋊C4 [×6], C22×C20 [×9], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4×C20 [×3], C5×C42⋊C2 [×3], D4×C20 [×6], Q8×C20 [×2], C10×C4○D4, C4○D4×C20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], C23 [×15], C10 [×15], C22×C4 [×14], C4○D4 [×4], C24, C20 [×8], C2×C10 [×35], C23×C4, C2×C4○D4 [×2], C2×C20 [×28], C22×C10 [×15], C4×C4○D4, C22×C20 [×14], C5×C4○D4 [×4], C23×C10, C23×C20, C10×C4○D4 [×2], C4○D4×C20

Smallest permutation representation of C4○D4×C20
On 160 points
Generators in S160
(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)
(1 101 59 29)(2 102 60 30)(3 103 41 31)(4 104 42 32)(5 105 43 33)(6 106 44 34)(7 107 45 35)(8 108 46 36)(9 109 47 37)(10 110 48 38)(11 111 49 39)(12 112 50 40)(13 113 51 21)(14 114 52 22)(15 115 53 23)(16 116 54 24)(17 117 55 25)(18 118 56 26)(19 119 57 27)(20 120 58 28)(61 122 156 93)(62 123 157 94)(63 124 158 95)(64 125 159 96)(65 126 160 97)(66 127 141 98)(67 128 142 99)(68 129 143 100)(69 130 144 81)(70 131 145 82)(71 132 146 83)(72 133 147 84)(73 134 148 85)(74 135 149 86)(75 136 150 87)(76 137 151 88)(77 138 152 89)(78 139 153 90)(79 140 154 91)(80 121 155 92)
(1 135 59 86)(2 136 60 87)(3 137 41 88)(4 138 42 89)(5 139 43 90)(6 140 44 91)(7 121 45 92)(8 122 46 93)(9 123 47 94)(10 124 48 95)(11 125 49 96)(12 126 50 97)(13 127 51 98)(14 128 52 99)(15 129 53 100)(16 130 54 81)(17 131 55 82)(18 132 56 83)(19 133 57 84)(20 134 58 85)(21 66 113 141)(22 67 114 142)(23 68 115 143)(24 69 116 144)(25 70 117 145)(26 71 118 146)(27 72 119 147)(28 73 120 148)(29 74 101 149)(30 75 102 150)(31 76 103 151)(32 77 104 152)(33 78 105 153)(34 79 106 154)(35 80 107 155)(36 61 108 156)(37 62 109 157)(38 63 110 158)(39 64 111 159)(40 65 112 160)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 141)(77 142)(78 143)(79 144)(80 145)(81 140)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 128)(90 129)(91 130)(92 131)(93 132)(94 133)(95 134)(96 135)(97 136)(98 137)(99 138)(100 139)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)

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

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

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

200 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4L 4M ··· 4AD 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AJ 20A ··· 20AV 20AW ··· 20DP order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C20 C4○D4 C5×C4○D4 kernel C4○D4×C20 C2×C4×C20 C5×C42⋊C2 D4×C20 Q8×C20 C10×C4○D4 C5×C4○D4 C4×C4○D4 C2×C42 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C20 C4 # reps 1 3 3 6 2 1 16 4 12 12 24 8 4 64 8 32

Matrix representation of C4○D4×C20 in GL3(𝔽41) generated by

 32 0 0 0 10 0 0 0 10
,
 1 0 0 0 32 0 0 0 32
,
 40 0 0 0 40 39 0 1 1
,
 1 0 0 0 1 0 0 40 40
G:=sub<GL(3,GF(41))| [32,0,0,0,10,0,0,0,10],[1,0,0,0,32,0,0,0,32],[40,0,0,0,40,1,0,39,1],[1,0,0,0,1,40,0,0,40] >;

C4○D4×C20 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{20}
% in TeX

G:=Group("C4oD4xC20");
// GroupNames label

G:=SmallGroup(320,1519);
// by ID

G=gap.SmallGroup(320,1519);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,856,304]);
// Polycyclic

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

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