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G = C8×D20order 320 = 26·5

Direct product of C8 and D20

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

Aliases: C8×D20, C4024D4, C42.256D10, C54(C8×D4), C42(C8×D5), (C4×C8)⋊3D5, (C4×C40)⋊19C2, C2010(C2×C8), D103(C2×C8), C2.1(C4×D20), C203C836C2, C10.34(C4×D4), C4.74(C2×D20), (C4×D20).29C2, (C2×D20).30C4, C20.294(C2×D4), (C2×C8).339D10, C4⋊Dic5.35C4, D101C841C2, C10.27(C8○D4), C10.28(C22×C8), D10⋊C4.28C4, C4.125(C4○D20), C20.241(C4○D4), (C4×C20).323C22, (C2×C40).342C22, (C2×C20).804C23, C2.2(D20.3C4), C2.6(D5×C2×C8), (D5×C2×C8)⋊11C2, C22.36(C2×C4×D5), (C2×C4).103(C4×D5), (C2×C20).393(C2×C4), (C2×C4×D5).336C22, (C2×Dic5).91(C2×C4), (C22×D5).68(C2×C4), (C2×C4).746(C22×D5), (C2×C10).160(C22×C4), (C2×C52C8).298C22, SmallGroup(320,313)

Series: Derived Chief Lower central Upper central

C1C10 — C8×D20
C1C5C10C20C2×C20C2×C4×D5C4×D20 — C8×D20
C5C10 — C8×D20
C1C2×C8C4×C8

Generators and relations for C8×D20
 G = < a,b,c | a8=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 446 in 134 conjugacy classes, 63 normal (33 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×6], C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×4], D10 [×4], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C52C8 [×2], C40 [×2], C40, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C8×D4, C8×D5 [×4], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20, C203C8, D101C8 [×2], C4×C40, C4×D20, D5×C2×C8 [×2], C8×D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C22×C8, C8○D4, C4×D5 [×2], D20 [×2], C22×D5, C8×D4, C8×D5 [×2], C2×C4×D5, C2×D20, C4○D20, C4×D20, D5×C2×C8, D20.3C4, C8×D20

Smallest permutation representation of C8×D20
On 160 points
Generators in S160
(1 141 36 134 99 61 102 41)(2 142 37 135 100 62 103 42)(3 143 38 136 81 63 104 43)(4 144 39 137 82 64 105 44)(5 145 40 138 83 65 106 45)(6 146 21 139 84 66 107 46)(7 147 22 140 85 67 108 47)(8 148 23 121 86 68 109 48)(9 149 24 122 87 69 110 49)(10 150 25 123 88 70 111 50)(11 151 26 124 89 71 112 51)(12 152 27 125 90 72 113 52)(13 153 28 126 91 73 114 53)(14 154 29 127 92 74 115 54)(15 155 30 128 93 75 116 55)(16 156 31 129 94 76 117 56)(17 157 32 130 95 77 118 57)(18 158 33 131 96 78 119 58)(19 159 34 132 97 79 120 59)(20 160 35 133 98 80 101 60)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(56 60)(57 59)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)(81 91)(82 90)(83 89)(84 88)(85 87)(92 100)(93 99)(94 98)(95 97)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)

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

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

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

104 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H8I8J8K8L8M···8T10A···10F20A···20X40A···40AF
order12222222444444444444558···888888···810···1020···2040···40
size1111101010101111222210101010221···1222210···102···22···22···2

104 irreducible representations

dim111111111122222222222
type+++++++++++
imageC1C2C2C2C2C2C4C4C4C8D4D5C4○D4D10D10C8○D4D20C4×D5C8×D5C4○D20D20.3C4
kernelC8×D20C203C8D101C8C4×C40C4×D20D5×C2×C8C4⋊Dic5D10⋊C4C2×D20D20C40C4×C8C20C42C2×C8C10C8C2×C4C4C4C2
# reps112112242162222448816816

Matrix representation of C8×D20 in GL3(𝔽41) generated by

300
030
003
,
4000
02839
0216
,
4000
0040
0400
G:=sub<GL(3,GF(41))| [3,0,0,0,3,0,0,0,3],[40,0,0,0,28,2,0,39,16],[40,0,0,0,0,40,0,40,0] >;

C8×D20 in GAP, Magma, Sage, TeX

C_8\times D_{20}
% in TeX

G:=Group("C8xD20");
// GroupNames label

G:=SmallGroup(320,313);
// by ID

G=gap.SmallGroup(320,313);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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