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G = C4.D40order 320 = 26·5

1st non-split extension by C4 of D40 acting via D40/D20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.1D8, C4.9D40, C42.5D10, C20.47SD16, C4⋊C82D5, C203C88C2, (C2×D20).2C4, C52(C4.D8), C4.20(D4⋊D5), (C2×C20).465D4, C204D4.5C2, (C2×C4).123D20, C4.6(C40⋊C2), C4.12(Q8⋊D5), (C4×C20).43C22, C2.5(D206C4), C2.4(D205C4), C10.27(D4⋊C4), C2.5(C20.46D4), C10.11(C4.D4), C22.62(D10⋊C4), (C5×C4⋊C8)⋊2C2, (C2×C4).16(C4×D5), (C2×C20).201(C2×C4), (C2×C4).229(C5⋊D4), (C2×C10).111(C22⋊C4), SmallGroup(320,43)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4.D40
C1C5C10C2×C10C2×C20C4×C20C204D4 — C4.D40
C5C2×C10C2×C20 — C4.D40
C1C22C42C4⋊C8

Generators and relations for C4.D40
 G = < a,b,c | a4=b40=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >

Subgroups: 542 in 84 conjugacy classes, 33 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4, C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], D4 [×6], C23 [×2], D5 [×2], C10 [×3], C42, C2×C8 [×2], C2×D4 [×4], C20 [×4], C20, D10 [×6], C2×C10, C4⋊C8, C4⋊C8, C41D4, C52C8, C40, D20 [×6], C2×C20 [×3], C22×D5 [×2], C4.D8, C2×C52C8, C4×C20, C2×C40, C2×D20 [×2], C2×D20 [×2], C203C8, C5×C4⋊C8, C204D4, C4.D40
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8 [×2], SD16 [×2], D10, C4.D4, D4⋊C4 [×2], C4×D5, D20, C5⋊D4, C4.D8, C40⋊C2, D40, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, D205C4, C20.46D4, C4.D40

Smallest permutation representation of C4.D40
On 160 points
Generators in S160
(1 61 146 105)(2 106 147 62)(3 63 148 107)(4 108 149 64)(5 65 150 109)(6 110 151 66)(7 67 152 111)(8 112 153 68)(9 69 154 113)(10 114 155 70)(11 71 156 115)(12 116 157 72)(13 73 158 117)(14 118 159 74)(15 75 160 119)(16 120 121 76)(17 77 122 81)(18 82 123 78)(19 79 124 83)(20 84 125 80)(21 41 126 85)(22 86 127 42)(23 43 128 87)(24 88 129 44)(25 45 130 89)(26 90 131 46)(27 47 132 91)(28 92 133 48)(29 49 134 93)(30 94 135 50)(31 51 136 95)(32 96 137 52)(33 53 138 97)(34 98 139 54)(35 55 140 99)(36 100 141 56)(37 57 142 101)(38 102 143 58)(39 59 144 103)(40 104 145 60)
(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)
(1 104 61 145 146 60 105 40)(2 39 106 59 147 144 62 103)(3 102 63 143 148 58 107 38)(4 37 108 57 149 142 64 101)(5 100 65 141 150 56 109 36)(6 35 110 55 151 140 66 99)(7 98 67 139 152 54 111 34)(8 33 112 53 153 138 68 97)(9 96 69 137 154 52 113 32)(10 31 114 51 155 136 70 95)(11 94 71 135 156 50 115 30)(12 29 116 49 157 134 72 93)(13 92 73 133 158 48 117 28)(14 27 118 47 159 132 74 91)(15 90 75 131 160 46 119 26)(16 25 120 45 121 130 76 89)(17 88 77 129 122 44 81 24)(18 23 82 43 123 128 78 87)(19 86 79 127 124 42 83 22)(20 21 84 41 125 126 80 85)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H20I···20P40A···40P
order12222244444558888888810···1020···2020···2040···40
size1111404022224224444202020202···22···24···44···4

59 irreducible representations

dim1111122222222224444
type++++++++++++++
imageC1C2C2C2C4D4D5D8SD16D10C4×D5D20C5⋊D4C40⋊C2D40C4.D4D4⋊D5Q8⋊D5C20.46D4
kernelC4.D40C203C8C5×C4⋊C8C204D4C2×D20C2×C20C4⋊C8C20C20C42C2×C4C2×C4C2×C4C4C4C10C4C4C2
# reps1111422442444881224

Matrix representation of C4.D40 in GL4(𝔽41) generated by

04000
1000
00400
00040
,
151500
152600
002818
002325
,
151500
261500
002818
002713
G:=sub<GL(4,GF(41))| [0,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[15,15,0,0,15,26,0,0,0,0,28,23,0,0,18,25],[15,26,0,0,15,15,0,0,0,0,28,27,0,0,18,13] >;

C4.D40 in GAP, Magma, Sage, TeX

C_4.D_{40}
% in TeX

G:=Group("C4.D40");
// GroupNames label

G:=SmallGroup(320,43);
// by ID

G=gap.SmallGroup(320,43);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,100,1123,794,136,12550]);
// Polycyclic

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

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