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G = C20.16D8order 320 = 26·5

16th non-split extension by C20 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.16D8, C20.15SD16, C42.217D10, C4.5(D4⋊D5), C10.56(C2×D8), C41D4.3D5, C53(C4.4D8), C202Q819C2, (C2×D4).53D10, (C2×C20).146D4, C4.3(D4.D5), C20.74(C4○D4), D4⋊Dic520C2, C10.57(C2×SD16), C4.22(D42D5), (C4×C20).118C22, (C2×C20).388C23, (D4×C10).69C22, C10.43(C4.4D4), C4⋊Dic5.154C22, C2.10(C20.17D4), (C4×C52C8)⋊14C2, C2.11(C2×D4⋊D5), (C5×C41D4).2C2, C2.11(C2×D4.D5), (C2×C10).519(C2×D4), (C2×C4).130(C5⋊D4), (C2×C4).486(C22×D5), C22.192(C2×C5⋊D4), (C2×C52C8).264C22, SmallGroup(320,697)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.16D8
C1C5C10C2×C10C2×C20C2×C52C8C4×C52C8 — C20.16D8
C5C10C2×C20 — C20.16D8
C1C22C42C41D4

Generators and relations for C20.16D8
 G = < a,b,c | a20=b8=1, c2=a10, bab-1=a9, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 414 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], C10 [×3], C10 [×2], C42, C4⋊C4 [×3], C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×6], C2×C10, C2×C10 [×6], C4×C8, D4⋊C4 [×4], C41D4, C4⋊Q8, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×3], C5×D4 [×8], C22×C10 [×2], C4.4D8, C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C2×Dic10, D4×C10 [×2], D4×C10 [×2], C4×C52C8, D4⋊Dic5 [×4], C202Q8, C5×C41D4, C20.16D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C2×D8, C2×SD16, C5⋊D4 [×2], C22×D5, C4.4D8, D4⋊D5 [×2], D4.D5 [×2], D42D5 [×2], C2×C5⋊D4, C2×D4⋊D5, C2×D4.D5, C20.17D4, C20.16D8

Smallest permutation representation of C20.16D8
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)
(1 147 60 96 28 140 107 62)(2 156 41 85 29 129 108 71)(3 145 42 94 30 138 109 80)(4 154 43 83 31 127 110 69)(5 143 44 92 32 136 111 78)(6 152 45 81 33 125 112 67)(7 141 46 90 34 134 113 76)(8 150 47 99 35 123 114 65)(9 159 48 88 36 132 115 74)(10 148 49 97 37 121 116 63)(11 157 50 86 38 130 117 72)(12 146 51 95 39 139 118 61)(13 155 52 84 40 128 119 70)(14 144 53 93 21 137 120 79)(15 153 54 82 22 126 101 68)(16 142 55 91 23 135 102 77)(17 151 56 100 24 124 103 66)(18 160 57 89 25 133 104 75)(19 149 58 98 26 122 105 64)(20 158 59 87 27 131 106 73)
(1 77 11 67)(2 76 12 66)(3 75 13 65)(4 74 14 64)(5 73 15 63)(6 72 16 62)(7 71 17 61)(8 70 18 80)(9 69 19 79)(10 68 20 78)(21 98 31 88)(22 97 32 87)(23 96 33 86)(24 95 34 85)(25 94 35 84)(26 93 36 83)(27 92 37 82)(28 91 38 81)(29 90 39 100)(30 89 40 99)(41 134 51 124)(42 133 52 123)(43 132 53 122)(44 131 54 121)(45 130 55 140)(46 129 56 139)(47 128 57 138)(48 127 58 137)(49 126 59 136)(50 125 60 135)(101 148 111 158)(102 147 112 157)(103 146 113 156)(104 145 114 155)(105 144 115 154)(106 143 116 153)(107 142 117 152)(108 141 118 151)(109 160 119 150)(110 159 120 149)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B8A···8H10A···10F10G···10N20A···20L
order1222224···444558···810···1010···1020···20
size1111882···240402210···102···28···84···4

50 irreducible representations

dim1111122222222444
type+++++++++++--
imageC1C2C2C2C2D4D5D8SD16C4○D4D10D10C5⋊D4D4⋊D5D4.D5D42D5
kernelC20.16D8C4×C52C8D4⋊Dic5C202Q8C5×C41D4C2×C20C41D4C20C20C20C42C2×D4C2×C4C4C4C4
# reps1141122444248444

Matrix representation of C20.16D8 in GL6(𝔽41)

4000000
0400000
0025000
0022300
0000040
000010
,
29290000
12290000
0043700
00143700
00001515
00002615
,
12120000
12290000
0037400
0027400
00001526
00002626

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,25,2,0,0,0,0,0,23,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[29,12,0,0,0,0,29,29,0,0,0,0,0,0,4,14,0,0,0,0,37,37,0,0,0,0,0,0,15,26,0,0,0,0,15,15],[12,12,0,0,0,0,12,29,0,0,0,0,0,0,37,27,0,0,0,0,4,4,0,0,0,0,0,0,15,26,0,0,0,0,26,26] >;

C20.16D8 in GAP, Magma, Sage, TeX

C_{20}._{16}D_8
% in TeX

G:=Group("C20.16D8");
// GroupNames label

G:=SmallGroup(320,697);
// by ID

G=gap.SmallGroup(320,697);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,590,135,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations

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