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G = C10×Q16order 160 = 25·5

Direct product of C10 and Q16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C10×Q16, C20.43D4, C20.46C23, C40.27C22, C4.8(C5×D4), C8.5(C2×C10), (C2×C8).4C10, (C2×C40).14C2, C10.76(C2×D4), C2.13(D4×C10), (C2×C10).54D4, Q8.1(C2×C10), (C2×Q8).4C10, (Q8×C10).9C2, C4.3(C22×C10), C22.16(C5×D4), (C5×Q8).12C22, (C2×C20).131C22, (C2×C4).27(C2×C10), SmallGroup(160,195)

Series: Derived Chief Lower central Upper central

C1C4 — C10×Q16
C1C2C4C20C5×Q8C5×Q16 — C10×Q16
C1C2C4 — C10×Q16
C1C2×C10C2×C20 — C10×Q16

Generators and relations for C10×Q16
 G = < a,b,c | a10=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 60 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×4], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×4], Q8 [×2], C10, C10 [×2], C2×C8, Q16 [×4], C2×Q8 [×2], C20 [×2], C20 [×4], C2×C10, C2×Q16, C40 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C2×C40, C5×Q16 [×4], Q8×C10 [×2], C10×Q16
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], Q16 [×2], C2×D4, C2×C10 [×7], C2×Q16, C5×D4 [×2], C22×C10, C5×Q16 [×2], D4×C10, C10×Q16

Smallest permutation representation of C10×Q16
Regular action on 160 points
Generators in S160
(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 23 50 154 61 15 58 37)(2 24 41 155 62 16 59 38)(3 25 42 156 63 17 60 39)(4 26 43 157 64 18 51 40)(5 27 44 158 65 19 52 31)(6 28 45 159 66 20 53 32)(7 29 46 160 67 11 54 33)(8 30 47 151 68 12 55 34)(9 21 48 152 69 13 56 35)(10 22 49 153 70 14 57 36)(71 149 87 111 91 127 109 131)(72 150 88 112 92 128 110 132)(73 141 89 113 93 129 101 133)(74 142 90 114 94 130 102 134)(75 143 81 115 95 121 103 135)(76 144 82 116 96 122 104 136)(77 145 83 117 97 123 105 137)(78 146 84 118 98 124 106 138)(79 147 85 119 99 125 107 139)(80 148 86 120 100 126 108 140)
(1 99 61 79)(2 100 62 80)(3 91 63 71)(4 92 64 72)(5 93 65 73)(6 94 66 74)(7 95 67 75)(8 96 68 76)(9 97 69 77)(10 98 70 78)(11 135 29 115)(12 136 30 116)(13 137 21 117)(14 138 22 118)(15 139 23 119)(16 140 24 120)(17 131 25 111)(18 132 26 112)(19 133 27 113)(20 134 28 114)(31 129 158 141)(32 130 159 142)(33 121 160 143)(34 122 151 144)(35 123 152 145)(36 124 153 146)(37 125 154 147)(38 126 155 148)(39 127 156 149)(40 128 157 150)(41 86 59 108)(42 87 60 109)(43 88 51 110)(44 89 52 101)(45 90 53 102)(46 81 54 103)(47 82 55 104)(48 83 56 105)(49 84 57 106)(50 85 58 107)

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

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

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

C10×Q16 is a maximal subgroup of
C40.15D4  Q16.Dic5  Q16.D10  C40.26D4  Dic53Q16  Q16⋊Dic5  (C2×Q16)⋊D5  D105Q16  D20.17D4  D103Q16  C40.36D4  C40.37D4  C40.28D4  C40.29D4  D20.30D4

70 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B5C5D8A8B8C8D10A···10L20A···20H20I···20X40A···40P
order12224444445555888810···1020···2020···2040···40
size1111224444111122221···12···24···42···2

70 irreducible representations

dim11111111222222
type++++++-
imageC1C2C2C2C5C10C10C10D4D4Q16C5×D4C5×D4C5×Q16
kernelC10×Q16C2×C40C5×Q16Q8×C10C2×Q16C2×C8Q16C2×Q8C20C2×C10C10C4C22C2
# reps1142441681144416

Matrix representation of C10×Q16 in GL4(𝔽41) generated by

31000
03100
00230
00023
,
333800
8800
002912
002929
,
1200
04000
00301
00111
G:=sub<GL(4,GF(41))| [31,0,0,0,0,31,0,0,0,0,23,0,0,0,0,23],[33,8,0,0,38,8,0,0,0,0,29,29,0,0,12,29],[1,0,0,0,2,40,0,0,0,0,30,1,0,0,1,11] >;

C10×Q16 in GAP, Magma, Sage, TeX

C_{10}\times Q_{16}
% in TeX

G:=Group("C10xQ16");
// GroupNames label

G:=SmallGroup(160,195);
// by ID

G=gap.SmallGroup(160,195);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,487,3604,1810,88]);
// Polycyclic

G:=Group<a,b,c|a^10=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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