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G = C42.143D10order 320 = 26·5

143rd non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.143D10, C10.1272+ 1+4, (C4×D20)⋊46C2, (D4×Dic5)⋊31C2, (Q8×Dic5)⋊20C2, C4.4D414D5, (C4×Dic10)⋊46C2, (C2×D4).176D10, C20⋊D4.11C2, (C2×Q8).139D10, C22⋊C4.36D10, Dic54D434C2, C20.126(C4○D4), C20.23D423C2, C4.16(D42D5), (C2×C20).505C23, (C4×C20).188C22, (C2×C10).225C24, D10.12D446C2, C2.51(D48D10), C23.47(C22×D5), Dic5.65(C4○D4), Dic5.5D441C2, (C2×D20).274C22, (D4×C10).158C22, C22.D2026C2, C4⋊Dic5.235C22, (C22×C10).55C23, (Q8×C10).129C22, (C22×D5).97C23, C22.246(C23×D5), C23.D5.58C22, D10⋊C4.37C22, C54(C22.53C24), (C4×Dic5).143C22, (C2×Dic5).266C23, (C2×Dic10).258C22, C10.D4.142C22, (C22×Dic5).145C22, C2.81(D5×C4○D4), C10.192(C2×C4○D4), C2.57(C2×D42D5), (C5×C4.4D4)⋊17C2, (C2×C4×D5).267C22, (C2×C4).198(C22×D5), (C2×C5⋊D4).63C22, (C5×C22⋊C4).67C22, SmallGroup(320,1353)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.143D10
C1C5C10C2×C10C22×D5C2×C4×D5D10.12D4 — C42.143D10
C5C2×C10 — C42.143D10
C1C22C4.4D4

Generators and relations for C42.143D10
 G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 854 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×2], C42, C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×6], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4, C4.4D4 [×3], C41D4, Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], C22.53C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic54D4 [×2], D10.12D4 [×2], Dic5.5D4 [×2], C22.D20 [×2], D4×Dic5, C20⋊D4, Q8×Dic5, C20.23D4, C5×C4.4D4, C42.143D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.53C24, D42D5 [×2], C23×D5, C2×D42D5, D5×C4○D4, D48D10, C42.143D10

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

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H···4O4P4Q5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order1222222244444444···4445510···101010101020···2020202020
size1111442020222244410···102020222···288884···48888

53 irreducible representations

dim11111111111122222224444
type++++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2D5C4○D4C4○D4D10D10D10D102+ 1+4D42D5D5×C4○D4D48D10
kernelC42.143D10C4×Dic10C4×D20Dic54D4D10.12D4Dic5.5D4C22.D20D4×Dic5C20⋊D4Q8×Dic5C20.23D4C5×C4.4D4C4.4D4Dic5C20C42C22⋊C4C2×D4C2×Q8C10C4C2C2
# reps11122221111124428221444

Matrix representation of C42.143D10 in GL6(𝔽41)

900000
090000
0040000
0004000
000090
00001632
,
32160000
3690000
001000
000100
000090
00001632
,
40200000
010000
006600
0035100
000028
00001539
,
40200000
410000
006600
0013500
00003933
0000162

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[32,36,0,0,0,0,16,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,0,32],[40,0,0,0,0,0,20,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,2,15,0,0,0,0,8,39],[40,4,0,0,0,0,20,1,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,39,16,0,0,0,0,33,2] >;

C42.143D10 in GAP, Magma, Sage, TeX

C_4^2._{143}D_{10}
% in TeX

G:=Group("C4^2.143D10");
// GroupNames label

G:=SmallGroup(320,1353);
// by ID

G=gap.SmallGroup(320,1353);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,297,80,12550]);
// Polycyclic

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

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