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G = C4×Dic10order 160 = 25·5

Direct product of C4 and Dic10

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

Aliases: C4×Dic10, C203Q8, C42.3D5, C52(C4×Q8), C4.9(C4×D5), (C4×C20).5C2, C10.1(C2×Q8), C20.40(C2×C4), (C2×C4).72D10, C10.1(C4○D4), C2.1(C4○D20), C4⋊Dic5.13C2, (C2×C10).9C23, Dic5.3(C2×C4), (C4×Dic5).9C2, C2.1(C2×Dic10), C10.14(C22×C4), (C2×C20).84C22, C10.D4.7C2, C22.8(C22×D5), (C2×Dic10).11C2, (C2×Dic5).26C22, C2.4(C2×C4×D5), SmallGroup(160,89)

Series: Derived Chief Lower central Upper central

C1C10 — C4×Dic10
C1C5C10C2×C10C2×Dic5C2×Dic10 — C4×Dic10
C5C10 — C4×Dic10
C1C2×C4C42

Generators and relations for C4×Dic10
 G = < a,b,c | a4=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 168 in 70 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×Q8, Dic10, C2×Dic5, C2×C20, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×Dic10, C4×Dic10
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, C22×C4, C2×Q8, C4○D4, D10, C4×Q8, Dic10, C4×D5, C22×D5, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic10

Smallest permutation representation of C4×Dic10
Regular action on 160 points
Generators in S160
(1 133 73 52)(2 134 74 53)(3 135 75 54)(4 136 76 55)(5 137 77 56)(6 138 78 57)(7 139 79 58)(8 140 80 59)(9 121 61 60)(10 122 62 41)(11 123 63 42)(12 124 64 43)(13 125 65 44)(14 126 66 45)(15 127 67 46)(16 128 68 47)(17 129 69 48)(18 130 70 49)(19 131 71 50)(20 132 72 51)(21 85 105 159)(22 86 106 160)(23 87 107 141)(24 88 108 142)(25 89 109 143)(26 90 110 144)(27 91 111 145)(28 92 112 146)(29 93 113 147)(30 94 114 148)(31 95 115 149)(32 96 116 150)(33 97 117 151)(34 98 118 152)(35 99 119 153)(36 100 120 154)(37 81 101 155)(38 82 102 156)(39 83 103 157)(40 84 104 158)
(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 35 11 25)(2 34 12 24)(3 33 13 23)(4 32 14 22)(5 31 15 21)(6 30 16 40)(7 29 17 39)(8 28 18 38)(9 27 19 37)(10 26 20 36)(41 144 51 154)(42 143 52 153)(43 142 53 152)(44 141 54 151)(45 160 55 150)(46 159 56 149)(47 158 57 148)(48 157 58 147)(49 156 59 146)(50 155 60 145)(61 111 71 101)(62 110 72 120)(63 109 73 119)(64 108 74 118)(65 107 75 117)(66 106 76 116)(67 105 77 115)(68 104 78 114)(69 103 79 113)(70 102 80 112)(81 121 91 131)(82 140 92 130)(83 139 93 129)(84 138 94 128)(85 137 95 127)(86 136 96 126)(87 135 97 125)(88 134 98 124)(89 133 99 123)(90 132 100 122)

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

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

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

C4×Dic10 is a maximal subgroup of
Dic103C8  Dic104C8  C4011Q8  C40⋊Q8  C42.16D10  Dic209C4  Dic5.5M4(2)  Dic10.3Q8  Dic105C8  C42.198D10  C42.36D10  Dic108D4  C4⋊Dic20  C20.7Q16  Dic104Q8  C42.51D10  C42.59D10  C42.61D10  Dic10.4Q8  Dic109D4  C20⋊Q16  Dic105Q8  Dic106Q8  C42.274D10  C42.277D10  C42.87D10  C42.88D10  C42.89D10  C42.91D10  C42.93D10  C42.96D10  C42.98D10  C42.99D10  C42.102D10  D45Dic10  C42.105D10  C42.106D10  D46Dic10  C42.108D10  Dic1023D4  Dic1024D4  C42.229D10  C42.114D10  C42.115D10  Dic1010Q8  C42.122D10  Q85Dic10  Q86Dic10  C4×Q8×D5  C42.125D10  C42.232D10  C42.134D10  C42.135D10  C42.136D10  C42.137D10  C42.139D10  Dic1010D4  C42.143D10  Dic107Q8  D207Q8  C42.152D10  C42.154D10  C42.159D10  C42.160D10  C42.162D10  C42.164D10  C42.166D10  Dic1011D4  Dic108Q8  Dic109Q8  D208Q8  D209Q8  C42.177D10  Dic35Dic10  Dic3014C4
C4×Dic10 is a maximal quotient of
(C2×C20)⋊Q8  C10.49(C4×D4)  C4⋊Dic515C4  C10.52(C4×D4)  C4011Q8  C40⋊Q8  C207(C4⋊C4)  (C2×C20)⋊10Q8  C10.92(C4×D4)  Dic35Dic10  Dic3014C4

52 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I···4P5A5B10A···10F20A···20X
order1222444444444···45510···1020···20
size11111111222210···10222···22···2

52 irreducible representations

dim11111112222222
type++++++-++-
imageC1C2C2C2C2C2C4Q8D5C4○D4D10Dic10C4×D5C4○D20
kernelC4×Dic10C4×Dic5C10.D4C4⋊Dic5C4×C20C2×Dic10Dic10C20C42C10C2×C4C4C4C2
# reps12211182226888

Matrix representation of C4×Dic10 in GL3(𝔽41) generated by

3200
090
009
,
4000
0911
03014
,
4000
02712
02814
G:=sub<GL(3,GF(41))| [32,0,0,0,9,0,0,0,9],[40,0,0,0,9,30,0,11,14],[40,0,0,0,27,28,0,12,14] >;

C4×Dic10 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{10}
% in TeX

G:=Group("C4xDic10");
// GroupNames label

G:=SmallGroup(160,89);
// by ID

G=gap.SmallGroup(160,89);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,217,103,50,4613]);
// Polycyclic

G:=Group<a,b,c|a^4=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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