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G = C2×D205C4order 320 = 26·5

Direct product of C2 and D205C4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D205C4, C23.58D20, C22.16D40, (C2×C8)⋊32D10, C2.3(C2×D40), (C22×C8)⋊4D5, D2027(C2×C4), (C2×D20)⋊19C4, (C22×C40)⋊3C2, (C2×C10).23D8, C10.16(C2×D8), (C2×C4).96D20, (C2×C40)⋊41C22, (C2×C20).474D4, C20.410(C2×D4), C103(D4⋊C4), C4⋊Dic547C22, C10.16(C2×SD16), (C2×C10).22SD16, (C22×D20).6C2, C22.53(C2×D20), C20.99(C22⋊C4), C20.171(C22×C4), (C2×C20).766C23, (C22×C10).138D4, (C22×C4).428D10, C4.27(D10⋊C4), (C2×D20).204C22, C22.12(C40⋊C2), (C22×C20).517C22, C22.49(D10⋊C4), C4.70(C2×C4×D5), C54(C2×D4⋊C4), C2.4(C2×C40⋊C2), (C2×C4⋊Dic5)⋊15C2, (C2×C4).116(C4×D5), C4.103(C2×C5⋊D4), (C2×C20).402(C2×C4), (C2×C10).156(C2×D4), C10.93(C2×C22⋊C4), C2.24(C2×D10⋊C4), (C2×C4).254(C5⋊D4), (C2×C4).714(C22×D5), (C2×C10).127(C22⋊C4), SmallGroup(320,739)

Series: Derived Chief Lower central Upper central

C1C20 — C2×D205C4
C1C5C10C20C2×C20C2×D20C22×D20 — C2×D205C4
C5C10C20 — C2×D205C4
C1C23C22×C4C22×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···2G2H2I2J2K4A4B4C4D4E4F4G4H5A5B8A···8H10A···10N20A···20P40A···40AF
order12···2222244444444558···810···1020···2040···40
size11···120202020222220202020222···22···22···22···2

92 irreducible representations

dim1111112222222222222
type++++++++++++++
imageC1C2C2C2C2C4D4D4D5D8SD16D10D10C4×D5D20C5⋊D4D20C40⋊C2D40
kernelC2×D205C4D205C4C2×C4⋊Dic5C22×C40C22×D20C2×D20C2×C20C22×C10C22×C8C2×C10C2×C10C2×C8C22×C4C2×C4C2×C4C2×C4C23C22C22
# reps141118312444284841616

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

40000
04000
00400
00040
,
1000
0100
001139
001627
,
1000
0100
00925
00532
,
32000
0100
00393
00402
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

׿
×
𝔽