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## G = C2×C4×D20order 320 = 26·5

### Direct product of C2×C4 and D20

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1518 in 426 conjugacy classes, 183 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×6], C22, C22 [×6], C22 [×32], C5, C2×C4 [×14], C2×C4 [×26], D4 [×16], C23, C23 [×20], D5 [×8], C10 [×3], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×18], C2×D4 [×12], C24 [×2], Dic5 [×4], C20 [×8], C20 [×2], D10 [×8], D10 [×24], C2×C10, C2×C10 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×D5 [×16], D20 [×16], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×14], C2×C20 [×2], C22×D5 [×12], C22×D5 [×8], C22×C10, C2×C4×D4, C4⋊Dic5 [×4], D10⋊C4 [×8], C4×C20 [×4], C2×C4×D5 [×8], C2×C4×D5 [×8], C2×D20 [×12], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], C4×D20 [×8], C2×C4⋊Dic5, C2×D10⋊C4 [×2], C2×C4×C20, D5×C22×C4 [×2], C22×D20, C2×C4×D20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C4×D5 [×4], D20 [×4], C22×D5 [×7], C2×C4×D4, C2×C4×D5 [×6], C2×D20 [×6], C4○D20 [×2], C23×D5, C4×D20 [×4], D5×C22×C4, C22×D20, C2×C4○D20, C2×C4×D20

Smallest permutation representation of C2×C4×D20
On 160 points
Generators in S160
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 100)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)
(1 36 101 150)(2 37 102 151)(3 38 103 152)(4 39 104 153)(5 40 105 154)(6 21 106 155)(7 22 107 156)(8 23 108 157)(9 24 109 158)(10 25 110 159)(11 26 111 160)(12 27 112 141)(13 28 113 142)(14 29 114 143)(15 30 115 144)(16 31 116 145)(17 32 117 146)(18 33 118 147)(19 34 119 148)(20 35 120 149)(41 122 79 95)(42 123 80 96)(43 124 61 97)(44 125 62 98)(45 126 63 99)(46 127 64 100)(47 128 65 81)(48 129 66 82)(49 130 67 83)(50 131 68 84)(51 132 69 85)(52 133 70 86)(53 134 71 87)(54 135 72 88)(55 136 73 89)(56 137 74 90)(57 138 75 91)(58 139 76 92)(59 140 77 93)(60 121 78 94)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 70)(20 69)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 100)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4P 4Q ··· 4X 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 ··· 10 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 D10 C4×D5 D20 C4○D20 kernel C2×C4×D20 C4×D20 C2×C4⋊Dic5 C2×D10⋊C4 C2×C4×C20 D5×C22×C4 C22×D20 C2×D20 C2×C20 C2×C42 C2×C10 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 8 1 2 1 2 1 16 4 2 4 8 6 16 16 16

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

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 9 0 0 0 0 0 0 9 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
,
 40 1 0 0 0 0 5 35 0 0 0 0 0 0 1 40 0 0 0 0 36 6 0 0 0 0 0 0 9 11 0 0 0 0 30 14
,
 1 0 0 0 0 0 36 40 0 0 0 0 0 0 40 0 0 0 0 0 5 1 0 0 0 0 0 0 9 11 0 0 0 0 30 32

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,9,30,0,0,0,0,11,32] >;

C2×C4×D20 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{20}
% in TeX

G:=Group("C2xC4xD20");
// GroupNames label

G:=SmallGroup(320,1145);
// by ID

G=gap.SmallGroup(320,1145);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,80,12550]);
// Polycyclic

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

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