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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 [×3], C4 [×4], C4 [×7], C22, C5, C2×C4 [×3], C2×C4 [×4], Q8 [×4], C10 [×3], C42, C42 [×2], C4⋊C4 [×3], C2×Q8, Dic5 [×4], Dic5 [×2], C20 [×4], C20, C2×C10, C4×Q8, Dic10 [×4], C2×Dic5 [×4], C2×C20 [×3], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4×C20, C2×Dic10, C4×Dic10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, C22×C4, C2×Q8, C4○D4, D10 [×3], C4×Q8, Dic10 [×2], C4×D5 [×2], 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 59 79 120)(2 60 80 101)(3 41 61 102)(4 42 62 103)(5 43 63 104)(6 44 64 105)(7 45 65 106)(8 46 66 107)(9 47 67 108)(10 48 68 109)(11 49 69 110)(12 50 70 111)(13 51 71 112)(14 52 72 113)(15 53 73 114)(16 54 74 115)(17 55 75 116)(18 56 76 117)(19 57 77 118)(20 58 78 119)(21 127 89 151)(22 128 90 152)(23 129 91 153)(24 130 92 154)(25 131 93 155)(26 132 94 156)(27 133 95 157)(28 134 96 158)(29 135 97 159)(30 136 98 160)(31 137 99 141)(32 138 100 142)(33 139 81 143)(34 140 82 144)(35 121 83 145)(36 122 84 146)(37 123 85 147)(38 124 86 148)(39 125 87 149)(40 126 88 150)
(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)
(1 141 11 151)(2 160 12 150)(3 159 13 149)(4 158 14 148)(5 157 15 147)(6 156 16 146)(7 155 17 145)(8 154 18 144)(9 153 19 143)(10 152 20 142)(21 59 31 49)(22 58 32 48)(23 57 33 47)(24 56 34 46)(25 55 35 45)(26 54 36 44)(27 53 37 43)(28 52 38 42)(29 51 39 41)(30 50 40 60)(61 135 71 125)(62 134 72 124)(63 133 73 123)(64 132 74 122)(65 131 75 121)(66 130 76 140)(67 129 77 139)(68 128 78 138)(69 127 79 137)(70 126 80 136)(81 108 91 118)(82 107 92 117)(83 106 93 116)(84 105 94 115)(85 104 95 114)(86 103 96 113)(87 102 97 112)(88 101 98 111)(89 120 99 110)(90 119 100 109)

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

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

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

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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