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G = C2×C4×Dic5order 160 = 25·5

Direct product of C2×C4 and Dic5

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

Aliases: C2×C4×Dic5, C102C42, C23.28D10, C53(C2×C42), C2011(C2×C4), (C2×C20)⋊12C4, (C2×C4).101D10, (C22×C4).10D5, C22.15(C4×D5), (C2×C10).40C23, (C22×C20).13C2, C10.35(C22×C4), C2.2(C22×Dic5), (C2×C20).113C22, C22.13(C2×Dic5), C22.19(C22×D5), (C22×C10).32C22, (C2×Dic5).64C22, (C22×Dic5).10C2, C2.3(C2×C4×D5), (C2×C10).53(C2×C4), SmallGroup(160,143)

Series: Derived Chief Lower central Upper central

C1C5 — C2×C4×Dic5
C1C5C10C2×C10C2×Dic5C22×Dic5 — C2×C4×Dic5
C5 — C2×C4×Dic5
C1C22×C4

Generators and relations for C2×C4×Dic5
 G = < a,b,c,d | a2=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 216 in 108 conjugacy classes, 81 normal (11 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×12], C23, C10, C10 [×6], C42 [×4], C22×C4, C22×C4 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×Dic5 [×12], C2×C20 [×6], C22×C10, C4×Dic5 [×4], C22×Dic5 [×2], C22×C20, C2×C4×Dic5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C2×C42, C4×D5 [×4], C2×Dic5 [×6], C22×D5, C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, C2×C4×Dic5

Smallest permutation representation of C2×C4×Dic5
Regular action on 160 points
Generators in S160
(1 68)(2 69)(3 70)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 127)(12 128)(13 129)(14 130)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 51)(30 52)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 81)(50 82)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)
(1 43 27 34)(2 44 28 35)(3 45 29 36)(4 46 30 37)(5 47 21 38)(6 48 22 39)(7 49 23 40)(8 50 24 31)(9 41 25 32)(10 42 26 33)(11 136 156 142)(12 137 157 143)(13 138 158 144)(14 139 159 145)(15 140 160 146)(16 131 151 147)(17 132 152 148)(18 133 153 149)(19 134 154 150)(20 135 155 141)(51 71 70 87)(52 72 61 88)(53 73 62 89)(54 74 63 90)(55 75 64 81)(56 76 65 82)(57 77 66 83)(58 78 67 84)(59 79 68 85)(60 80 69 86)(91 111 107 127)(92 112 108 128)(93 113 109 129)(94 114 110 130)(95 115 101 121)(96 116 102 122)(97 117 103 123)(98 118 104 124)(99 119 105 125)(100 120 106 126)
(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)
(1 151 6 156)(2 160 7 155)(3 159 8 154)(4 158 9 153)(5 157 10 152)(11 27 16 22)(12 26 17 21)(13 25 18 30)(14 24 19 29)(15 23 20 28)(31 134 36 139)(32 133 37 138)(33 132 38 137)(34 131 39 136)(35 140 40 135)(41 149 46 144)(42 148 47 143)(43 147 48 142)(44 146 49 141)(45 145 50 150)(51 130 56 125)(52 129 57 124)(53 128 58 123)(54 127 59 122)(55 126 60 121)(61 113 66 118)(62 112 67 117)(63 111 68 116)(64 120 69 115)(65 119 70 114)(71 94 76 99)(72 93 77 98)(73 92 78 97)(74 91 79 96)(75 100 80 95)(81 106 86 101)(82 105 87 110)(83 104 88 109)(84 103 89 108)(85 102 90 107)

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

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

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

C2×C4×Dic5 is a maximal subgroup of
C20.32C42  (C2×C40)⋊15C4  C20.33C42  C10.(C4⋊C8)  (C2×C20)⋊Q8  C10.49(C4×D4)  Dic5.15C42  Dic52C42  C52(C428C4)  C52(C425C4)  C10.51(C4×D4)  C2.(C4×D20)  C4⋊Dic515C4  C10.52(C4×D4)  D102C42  C10.54(C4×D4)  C10.55(C4×D4)  Dic5.14M4(2)  Dic5.9M4(2)  C424Dic5  C24.3D10  C24.4D10  C24.8D10  C24.13D10  C10.96(C4×D4)  C204(C4⋊C4)  (C2×Dic5)⋊6Q8  C205(C4⋊C4)  C20.48(C4⋊C4)  C10.97(C4×D4)  C4⋊C45Dic5  C206(C4⋊C4)  (C2×D20)⋊22C4  C10.90(C4×D4)  Dic55M4(2)  C24.19D10  (Q8×C10)⋊17C4  Dic5.12M4(2)  C20.34M4(2)  Dic5.13M4(2)  C208M4(2)  C20.30M4(2)  D5×C2×C42  C42.88D10  C42.188D10  C42.102D10  C20⋊(C4○D4)  C4⋊C4.178D10  (Q8×Dic5)⋊C2  C22⋊Q825D5  C4⋊C4.197D10  (C2×C20)⋊17D4
C2×C4×Dic5 is a maximal quotient of
C424Dic5  C20.35C42  C20.42C42  C20.37C42

64 conjugacy classes

class 1 2A···2G4A···4H4I···4X5A5B10A···10N20A···20P
order12···24···44···45510···1020···20
size11···11···15···5222···22···2

64 irreducible representations

dim11111122222
type+++++-++
imageC1C2C2C2C4C4D5Dic5D10D10C4×D5
kernelC2×C4×Dic5C4×Dic5C22×Dic5C22×C20C2×Dic5C2×C20C22×C4C2×C4C2×C4C23C22
# reps1421168284216

Matrix representation of C2×C4×Dic5 in GL4(𝔽41) generated by

1000
04000
00400
00040
,
9000
0100
0090
0009
,
1000
0100
00140
00366
,
40000
04000
002832
002813
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,1,36,0,0,40,6],[40,0,0,0,0,40,0,0,0,0,28,28,0,0,32,13] >;

C2×C4×Dic5 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm Dic}_5
% in TeX

G:=Group("C2xC4xDic5");
// GroupNames label

G:=SmallGroup(160,143);
// by ID

G=gap.SmallGroup(160,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,86,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽