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G = C40.8D4order 320 = 26·5

8th non-split extension by C40 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.8D4, C20.49D8, C8.18D20, Dic20.6C4, C8.3(C4×D5), C40.64(C2×C4), (C2×C20).96D4, (C2×C8).44D10, C4.22(D4⋊D5), C52(C8.17D4), (C2×C10).5SD16, C8.C4.3D5, C20.4C8.3C2, C22.4(Q8⋊D5), C4.5(D10⋊C4), C20.52(C22⋊C4), (C2×C40).101C22, (C2×Dic20).12C2, C2.10(D206C4), C10.23(D4⋊C4), (C5×C8.C4).2C2, (C2×C4).19(C5⋊D4), SmallGroup(320,54)

Series: Derived Chief Lower central Upper central

C1C40 — C40.8D4
C1C5C10C20C2×C20C2×C40C2×Dic20 — C40.8D4
C5C10C20C40 — C40.8D4
C1C2C2×C4C2×C8C8.C4

Generators and relations for C40.8D4
 G = < a,b,c | a40=1, b4=a20, c2=a5, bab-1=a31, cac-1=a9, cbc-1=a5b3 >

2C2
20C4
20C4
2C10
4C8
10Q8
10Q8
20C2×C4
20Q8
4Dic5
4Dic5
2M4(2)
5Q16
5Q16
10C16
10Q16
10C2×Q8
2Dic10
2Dic10
4Dic10
4C40
4C2×Dic5
5M5(2)
5C2×Q16
2Dic20
2C52C16
2C5×M4(2)
2C2×Dic10
5C8.17D4

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

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

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

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

44 conjugacy classes

class 1 2A2B4A4B4C4D5A5B8A8B8C8D8E10A10B10C10D16A16B16C16D20A20B20C20D20E20F40A···40H40I···40P
order12244445588888101010101616161620202020202040···4040···40
size11222404022224882244202020202222444···48···8

44 irreducible representations

dim111112222222224444
type++++++++++-++-
imageC1C2C2C2C4D4D4D5D8SD16D10C4×D5D20C5⋊D4C8.17D4D4⋊D5Q8⋊D5C40.8D4
kernelC40.8D4C20.4C8C5×C8.C4C2×Dic20Dic20C40C2×C20C8.C4C20C2×C10C2×C8C8C8C2×C4C5C4C22C1
# reps111141122224442228

Matrix representation of C40.8D4 in GL4(𝔽241) generated by

1851400
21820800
0021447
0018133
,
0010
0001
4415600
8819700
,
0020786
005134
7317600
8216800
G:=sub<GL(4,GF(241))| [185,218,0,0,14,208,0,0,0,0,214,181,0,0,47,33],[0,0,44,88,0,0,156,197,1,0,0,0,0,1,0,0],[0,0,73,82,0,0,176,168,207,51,0,0,86,34,0,0] >;

C40.8D4 in GAP, Magma, Sage, TeX

C_{40}._8D_4
% in TeX

G:=Group("C40.8D4");
// GroupNames label

G:=SmallGroup(320,54);
// by ID

G=gap.SmallGroup(320,54);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,758,184,675,794,192,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=1,b^4=a^20,c^2=a^5,b*a*b^-1=a^31,c*a*c^-1=a^9,c*b*c^-1=a^5*b^3>;
// generators/relations

Export

Subgroup lattice of C40.8D4 in TeX

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