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G = C42.16D10order 320 = 26·5

16th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.16D10, C81(C4×D5), C4011(C2×C4), C40⋊C25C4, C8⋊C41D5, (C4×D20).5C2, C405C416C2, C2.14(C4×D20), C10.41(C4×D4), (C2×C8).53D10, (C4×Dic10)⋊2C2, D20.28(C2×C4), (C2×C20).236D4, (C2×C4).114D20, Dic1018(C2×C4), C52(SD16⋊C4), C2.1(C8⋊D10), C10.2(C8⋊C22), (C2×C40).54C22, (C4×C20).14C22, C22.30(C2×D20), D205C4.15C2, C4.106(C4○D20), C20.222(C4○D4), C20.44D437C2, C20.164(C22×C4), (C2×C20).731C23, C2.1(C8.D10), C10.3(C8.C22), (C2×D20).195C22, C4⋊Dic5.265C22, (C2×Dic10).214C22, C4.63(C2×C4×D5), (C5×C8⋊C4)⋊2C2, (C2×C40⋊C2).1C2, (C2×C10).114(C2×D4), (C2×C4).675(C22×D5), SmallGroup(320,337)

Series: Derived Chief Lower central Upper central

C1C20 — C42.16D10
C1C5C10C2×C10C2×C20C2×D20C2×C40⋊C2 — C42.16D10
C5C10C20 — C42.16D10
C1C22C42C8⋊C4

Generators and relations for C42.16D10
 G = < a,b,c,d | a4=b4=1, c10=b, d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=bc9 >

Subgroups: 518 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×3], C23, D5 [×2], C10 [×3], C42, C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C40 [×2], C40, Dic10 [×2], Dic10, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×3], C2×C20 [×3], C22×D5, SD16⋊C4, C40⋊C2 [×4], C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C4×C20, C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, C405C4, D205C4, C5×C8⋊C4, C4×Dic10, C4×D20, C2×C40⋊C2, C42.16D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22, C8.C22, C4×D5 [×2], D20 [×2], C22×D5, SD16⋊C4, C2×C4×D5, C2×D20, C4○D20, C4×D20, C8⋊D10, C8.D10, C42.16D10

Smallest permutation representation of C42.16D10
On 160 points
Generators in S160
(1 87 55 159)(2 108 56 140)(3 89 57 121)(4 110 58 142)(5 91 59 123)(6 112 60 144)(7 93 61 125)(8 114 62 146)(9 95 63 127)(10 116 64 148)(11 97 65 129)(12 118 66 150)(13 99 67 131)(14 120 68 152)(15 101 69 133)(16 82 70 154)(17 103 71 135)(18 84 72 156)(19 105 73 137)(20 86 74 158)(21 107 75 139)(22 88 76 160)(23 109 77 141)(24 90 78 122)(25 111 79 143)(26 92 80 124)(27 113 41 145)(28 94 42 126)(29 115 43 147)(30 96 44 128)(31 117 45 149)(32 98 46 130)(33 119 47 151)(34 100 48 132)(35 81 49 153)(36 102 50 134)(37 83 51 155)(38 104 52 136)(39 85 53 157)(40 106 54 138)
(1 11 21 31)(2 12 22 32)(3 13 23 33)(4 14 24 34)(5 15 25 35)(6 16 26 36)(7 17 27 37)(8 18 28 38)(9 19 29 39)(10 20 30 40)(41 51 61 71)(42 52 62 72)(43 53 63 73)(44 54 64 74)(45 55 65 75)(46 56 66 76)(47 57 67 77)(48 58 68 78)(49 59 69 79)(50 60 70 80)(81 91 101 111)(82 92 102 112)(83 93 103 113)(84 94 104 114)(85 95 105 115)(86 96 106 116)(87 97 107 117)(88 98 108 118)(89 99 109 119)(90 100 110 120)(121 131 141 151)(122 132 142 152)(123 133 143 153)(124 134 144 154)(125 135 145 155)(126 136 146 156)(127 137 147 157)(128 138 148 158)(129 139 149 159)(130 140 150 160)
(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 10 21 30)(2 29 22 9)(3 8 23 28)(4 27 24 7)(5 6 25 26)(11 40 31 20)(12 19 32 39)(13 38 33 18)(14 17 34 37)(15 36 35 16)(41 78 61 58)(42 57 62 77)(43 76 63 56)(44 55 64 75)(45 74 65 54)(46 53 66 73)(47 72 67 52)(48 51 68 71)(49 70 69 50)(59 60 79 80)(81 102 101 82)(83 100 103 120)(84 119 104 99)(85 98 105 118)(86 117 106 97)(87 96 107 116)(88 115 108 95)(89 94 109 114)(90 113 110 93)(91 92 111 112)(121 126 141 146)(122 145 142 125)(123 124 143 144)(127 160 147 140)(128 139 148 159)(129 158 149 138)(130 137 150 157)(131 156 151 136)(132 135 152 155)(133 154 153 134)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G···4L5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order1222224···44···455888810···1020···2020···2040···40
size111120202···220···202244442···22···24···44···4

62 irreducible representations

dim111111111222222224444
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10C4×D5D20C4○D20C8⋊C22C8.C22C8⋊D10C8.D10
kernelC42.16D10C20.44D4C405C4D205C4C5×C8⋊C4C4×Dic10C4×D20C2×C40⋊C2C40⋊C2C2×C20C8⋊C4C20C42C2×C8C8C2×C4C4C10C10C2C2
# reps111111118222248881144

Matrix representation of C42.16D10 in GL6(𝔽41)

900000
090000
00184390
003731039
0011162337
002522410
,
100000
010000
00303200
0091100
00003032
0000911
,
0340000
6350000
001710512
003129293
0013172431
0024171012
,
6340000
5350000
002639125
002115329
007203124
0012341210

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,18,37,11,25,0,0,4,31,16,22,0,0,39,0,23,4,0,0,0,39,37,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,9,0,0,0,0,32,11,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[0,6,0,0,0,0,34,35,0,0,0,0,0,0,17,31,13,24,0,0,10,29,17,17,0,0,5,29,24,10,0,0,12,3,31,12],[6,5,0,0,0,0,34,35,0,0,0,0,0,0,26,21,7,12,0,0,39,15,20,34,0,0,12,3,31,12,0,0,5,29,24,10] >;

C42.16D10 in GAP, Magma, Sage, TeX

C_4^2._{16}D_{10}
% in TeX

G:=Group("C4^2.16D10");
// GroupNames label

G:=SmallGroup(320,337);
// by ID

G=gap.SmallGroup(320,337);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,387,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=b,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;
// generators/relations

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