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G = C40.43D4order 320 = 26·5

43rd non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.43D4, C4.24(D4×D5), (C2×SD16)⋊9D5, C52C8.32D4, (C10×SD16)⋊7C2, (C8×Dic5)⋊10C2, (C2×D4).71D10, C20.175(C2×D4), (C2×C8).262D10, C54(C8.12D4), C8.20(C5⋊D4), (C2×Q8).53D10, C10.62(C4○D8), C20.23D44C2, C20.17D46C2, C22.265(D4×D5), C10.30(C41D4), C2.21(C20⋊D4), (C2×C20).445C23, (C2×C40).163C22, (C2×Dic5).158D4, (D4×C10).94C22, (Q8×C10).75C22, (C2×D20).123C22, C2.28(SD163D5), (C4×Dic5).272C22, (C2×Dic10).130C22, C4.8(C2×C5⋊D4), (C2×D4⋊D5).9C2, (C2×C40⋊C2)⋊29C2, (C2×C5⋊Q16)⋊18C2, (C2×C10).357(C2×D4), (C2×C4).534(C22×D5), (C2×C52C8).281C22, SmallGroup(320,795)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C40.43D4
C1C5C10C20C2×C20C4×Dic5C20.17D4 — C40.43D4
C5C10C2×C20 — C40.43D4
C1C22C2×C4C2×SD16

Generators and relations for C40.43D4
 G = < a,b,c | a40=b4=1, c2=a20, bab-1=a9, cac-1=a19, cbc-1=a20b-1 >

Subgroups: 558 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], D5, C10, C10 [×2], C10, C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16, C2×SD16, C2×Q16, C52C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C8.12D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, D10⋊C4 [×2], D4⋊D5 [×2], C5⋊Q16 [×2], C23.D5 [×2], C2×C40, C5×SD16 [×2], C2×Dic10, C2×D20, D4×C10, Q8×C10, C8×Dic5, C2×C40⋊C2, C2×D4⋊D5, C20.17D4, C2×C5⋊Q16, C20.23D4, C10×SD16, C40.43D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C4○D8 [×2], C5⋊D4 [×2], C22×D5, C8.12D4, D4×D5 [×2], C2×C5⋊D4, SD163D5 [×2], C20⋊D4, C40.43D4

Smallest permutation representation of C40.43D4
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 79 107 135)(2 48 108 144)(3 57 109 153)(4 66 110 122)(5 75 111 131)(6 44 112 140)(7 53 113 149)(8 62 114 158)(9 71 115 127)(10 80 116 136)(11 49 117 145)(12 58 118 154)(13 67 119 123)(14 76 120 132)(15 45 81 141)(16 54 82 150)(17 63 83 159)(18 72 84 128)(19 41 85 137)(20 50 86 146)(21 59 87 155)(22 68 88 124)(23 77 89 133)(24 46 90 142)(25 55 91 151)(26 64 92 160)(27 73 93 129)(28 42 94 138)(29 51 95 147)(30 60 96 156)(31 69 97 125)(32 78 98 134)(33 47 99 143)(34 56 100 152)(35 65 101 121)(36 74 102 130)(37 43 103 139)(38 52 104 148)(39 61 105 157)(40 70 106 126)
(1 16 21 36)(2 35 22 15)(3 14 23 34)(4 33 24 13)(5 12 25 32)(6 31 26 11)(7 10 27 30)(8 29 28 9)(17 40 37 20)(18 19 38 39)(41 128 61 148)(42 147 62 127)(43 126 63 146)(44 145 64 125)(45 124 65 144)(46 143 66 123)(47 122 67 142)(48 141 68 121)(49 160 69 140)(50 139 70 159)(51 158 71 138)(52 137 72 157)(53 156 73 136)(54 135 74 155)(55 154 75 134)(56 133 76 153)(57 152 77 132)(58 131 78 151)(59 150 79 130)(60 129 80 149)(81 108 101 88)(82 87 102 107)(83 106 103 86)(84 85 104 105)(89 100 109 120)(90 119 110 99)(91 98 111 118)(92 117 112 97)(93 96 113 116)(94 115 114 95)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444558888888810···1010101010202020202020202040···40
size11118402281010101040222222101010102···28888444488884···4

50 irreducible representations

dim11111111222222222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5D10D10D10C4○D8C5⋊D4D4×D5D4×D5SD163D5
kernelC40.43D4C8×Dic5C2×C40⋊C2C2×D4⋊D5C20.17D4C2×C5⋊Q16C20.23D4C10×SD16C52C8C40C2×Dic5C2×SD16C2×C8C2×D4C2×Q8C10C8C4C22C2
# reps11111111222222288228

Matrix representation of C40.43D4 in GL6(𝔽41)

25360000
35160000
0035700
0035000
00003030
0000260
,
25360000
35160000
00354000
0035600
0000320
0000032
,
25360000
10160000
00354000
0035600
00001111
00001530

G:=sub<GL(6,GF(41))| [25,35,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,7,0,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[25,35,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[25,10,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,11,15,0,0,0,0,11,30] >;

C40.43D4 in GAP, Magma, Sage, TeX

C_{40}._{43}D_4
% in TeX

G:=Group("C40.43D4");
// GroupNames label

G:=SmallGroup(320,795);
// by ID

G=gap.SmallGroup(320,795);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^19,c*b*c^-1=a^20*b^-1>;
// generators/relations

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