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G = C20.9D8order 320 = 26·5

9th non-split extension by C20 of D8 acting via D8/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.9D8, C20.8SD16, C42.9D10, C203C810C2, C41D4.1D5, C53(C4.D8), C4.12(D4⋊D5), (D4×C10).16C4, (C2×C20).106D4, C4.6(D4.D5), (C2×D4).3Dic5, (C4×C20).47C22, C2.4(C20.D4), C2.4(D4⋊Dic5), C10.39(D4⋊C4), C10.16(C4.D4), C22.41(C23.D5), (C5×C41D4).1C2, (C2×C20).344(C2×C4), (C2×C4).11(C2×Dic5), (C2×C4).176(C5⋊D4), (C2×C10).166(C22⋊C4), SmallGroup(320,102)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C20.9D8
C1C5C10C2×C10C2×C20C4×C20C203C8 — C20.9D8
C5C2×C10C2×C20 — C20.9D8
C1C22C42C41D4

Generators and relations for C20.9D8
 G = < a,b,c | a20=b8=1, c2=a5, bab-1=a-1, cac-1=a9, cbc-1=a5b-1 >

Subgroups: 270 in 84 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4, C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C10, C10 [×2], C10 [×2], C42, C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C20 [×4], C20, C2×C10, C2×C10 [×6], C4⋊C8 [×2], C41D4, C52C8 [×2], C2×C20, C2×C20 [×2], C5×D4 [×6], C22×C10 [×2], C4.D8, C2×C52C8 [×2], C4×C20, D4×C10 [×2], D4×C10 [×2], C203C8 [×2], C5×C41D4, C20.9D8
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8 [×2], SD16 [×2], Dic5 [×2], D10, C4.D4, D4⋊C4 [×2], C2×Dic5, C5⋊D4 [×2], C4.D8, D4⋊D5 [×2], D4.D5 [×2], C23.D5, D4⋊Dic5 [×2], C20.D4, C20.9D8

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E5A5B8A···8H10A···10F10G···10N20A···20L
order12222244444558···810···1010···1020···20
size111188222242220···202···28···84···4

47 irreducible representations

dim111122222224444
type+++++++-++-
imageC1C2C2C4D4D5D8SD16D10Dic5C5⋊D4C4.D4D4⋊D5D4.D5C20.D4
kernelC20.9D8C203C8C5×C41D4D4×C10C2×C20C41D4C20C20C42C2×D4C2×C4C10C4C4C2
# reps121422442481444

Matrix representation of C20.9D8 in GL6(𝔽41)

010000
4000000
007100
0040000
0000400
0000040
,
26150000
15150000
00291400
00161200
00001515
00002615
,
26150000
26260000
00291400
00161200
00001515
00001526

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,7,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[26,15,0,0,0,0,15,15,0,0,0,0,0,0,29,16,0,0,0,0,14,12,0,0,0,0,0,0,15,26,0,0,0,0,15,15],[26,26,0,0,0,0,15,26,0,0,0,0,0,0,29,16,0,0,0,0,14,12,0,0,0,0,0,0,15,15,0,0,0,0,15,26] >;

C20.9D8 in GAP, Magma, Sage, TeX

C_{20}._9D_8
% in TeX

G:=Group("C20.9D8");
// GroupNames label

G:=SmallGroup(320,102);
// by ID

G=gap.SmallGroup(320,102);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,100,1571,570,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^5*b^-1>;
// generators/relations

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