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## G = C2×D20⋊5C4order 320 = 26·5

### Direct product of C2 and D20⋊5C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D20⋊5C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C22×D20 — C2×D20⋊5C4
 Lower central C5 — C10 — C20 — C2×D20⋊5C4
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×D205C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b3c >

Subgroups: 1054 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×10], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×9], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×6], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C40 [×2], D20 [×4], D20 [×6], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×10], C22×C10, C2×D4⋊C4, C4⋊Dic5 [×2], C4⋊Dic5, C2×C40 [×2], C2×C40 [×2], C2×D20 [×6], C2×D20 [×3], C22×Dic5, C22×C20, C23×D5, D205C4 [×4], C2×C4⋊Dic5, C22×C40, C22×D20, C2×D205C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D10 [×3], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×D4⋊C4, C40⋊C2 [×2], D40 [×2], D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D205C4 [×4], C2×C40⋊C2, C2×D40, C2×D10⋊C4, C2×D205C4

Smallest permutation representation of C2×D205C4
On 160 points
Generators in S160
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 101)(19 102)(20 103)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(121 158)(122 159)(123 160)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(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)(2 102)(3 101)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 105)(20 104)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)
(1 152 91 49)(2 151 92 48)(3 150 93 47)(4 149 94 46)(5 148 95 45)(6 147 96 44)(7 146 97 43)(8 145 98 42)(9 144 99 41)(10 143 100 60)(11 142 81 59)(12 141 82 58)(13 160 83 57)(14 159 84 56)(15 158 85 55)(16 157 86 54)(17 156 87 53)(18 155 88 52)(19 154 89 51)(20 153 90 50)(21 102 137 78)(22 101 138 77)(23 120 139 76)(24 119 140 75)(25 118 121 74)(26 117 122 73)(27 116 123 72)(28 115 124 71)(29 114 125 70)(30 113 126 69)(31 112 127 68)(32 111 128 67)(33 110 129 66)(34 109 130 65)(35 108 131 64)(36 107 132 63)(37 106 133 62)(38 105 134 61)(39 104 135 80)(40 103 136 79)

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 20 20 20 20 2 2 2 2 20 20 20 20 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 D10 C4×D5 D20 C5⋊D4 D20 C40⋊C2 D40 kernel C2×D20⋊5C4 D20⋊5C4 C2×C4⋊Dic5 C22×C40 C22×D20 C2×D20 C2×C20 C22×C10 C22×C8 C2×C10 C2×C10 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 3 1 2 4 4 4 2 8 4 8 4 16 16

Matrix representation of C2×D205C4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 11 39 0 0 16 27
,
 1 0 0 0 0 1 0 0 0 0 9 25 0 0 5 32
,
 32 0 0 0 0 1 0 0 0 0 39 3 0 0 40 2
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,11,16,0,0,39,27],[1,0,0,0,0,1,0,0,0,0,9,5,0,0,25,32],[32,0,0,0,0,1,0,0,0,0,39,40,0,0,3,2] >;

C2×D205C4 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes_5C_4
% in TeX

G:=Group("C2xD20:5C4");
// GroupNames label

G:=SmallGroup(320,739);
// by ID

G=gap.SmallGroup(320,739);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,142,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^3*c>;
// generators/relations

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