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G = D20⋊C8order 320 = 26·5

1st semidirect product of D20 and C8 acting via C8/C2=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D201C8, Dic5.20D8, C20.1M4(2), Dic5.15SD16, C51(D4⋊C8), C10.8C4≀C2, C4⋊C4.3F5, C20.1(C2×C8), C20⋊C81C2, C4.1(D5⋊C8), C4.1(C4.F5), (C2×D20).10C4, C10.1(C22⋊C8), C2.1(D20⋊C4), C2.1(Q82F5), (C2×Dic5).95D4, D208C4.15C2, C10.2(D4⋊C4), C2.3(D10⋊C8), C22.31(C22⋊F5), (C4×Dic5).186C22, (C4×C5⋊C8)⋊1C2, (C5×C4⋊C4).3C4, (C2×C4).64(C2×F5), (C2×C20).30(C2×C4), (C2×C10).21(C22⋊C4), SmallGroup(320,209)

Series: Derived Chief Lower central Upper central

C1C20 — D20⋊C8
C1C5C10C2×C10C2×Dic5C4×Dic5C20⋊C8 — D20⋊C8
C5C10C20 — D20⋊C8
C1C22C2×C4C4⋊C4

Generators and relations for D20⋊C8
 G = < a,b,c | a20=b2=c8=1, bab=a-1, cac-1=a3, cbc-1=a17b >

Subgroups: 402 in 82 conjugacy classes, 30 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×3], C2×C4, C2×C4 [×5], D4 [×3], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C22×C4, C2×D4, Dic5 [×2], Dic5, C20 [×2], C20, D10 [×4], C2×C10, C4×C8, C4⋊C8, C4×D4, C5⋊C8 [×3], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×C20, C2×C20, C22×D5, D4⋊C8, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×C5⋊C8 [×2], C2×C4×D5, C2×D20, C4×C5⋊C8, C20⋊C8, D208C4, D20⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), D8, SD16, F5, C22⋊C8, D4⋊C4, C4≀C2, C2×F5, D4⋊C8, D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, D20⋊C4, Q82F5, D20⋊C8

Smallest permutation representation of D20⋊C8
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 44)(42 43)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 144)(142 143)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)
(1 137 43 31 119 75 143 84)(2 124 52 34 120 62 152 87)(3 131 41 37 101 69 141 90)(4 138 50 40 102 76 150 93)(5 125 59 23 103 63 159 96)(6 132 48 26 104 70 148 99)(7 139 57 29 105 77 157 82)(8 126 46 32 106 64 146 85)(9 133 55 35 107 71 155 88)(10 140 44 38 108 78 144 91)(11 127 53 21 109 65 153 94)(12 134 42 24 110 72 142 97)(13 121 51 27 111 79 151 100)(14 128 60 30 112 66 160 83)(15 135 49 33 113 73 149 86)(16 122 58 36 114 80 158 89)(17 129 47 39 115 67 147 92)(18 136 56 22 116 74 156 95)(19 123 45 25 117 61 145 98)(20 130 54 28 118 68 154 81)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122222444444444458···8888810101020···20
size11112020224455551010410···10202020204448···8

38 irreducible representations

dim1111111222224444488
type+++++++++++
imageC1C2C2C2C4C4C8D4D8SD16M4(2)C4≀C2F5C2×F5D5⋊C8C4.F5C22⋊F5D20⋊C4Q82F5
kernelD20⋊C8C4×C5⋊C8C20⋊C8D208C4C5×C4⋊C4C2×D20D20C2×Dic5Dic5Dic5C20C10C4⋊C4C2×C4C4C4C22C2C2
# reps1111228222241122211

Matrix representation of D20⋊C8 in GL8(𝔽41)

400000000
040000000
00010000
004000000
000040100
000040010
000040001
000040000
,
400000000
91000000
00010000
00100000
000040000
000040001
000040010
000040100
,
2132000000
720000000
0026150000
0015150000
0000932230
0000323209
0000903232
0000023329

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[40,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[21,7,0,0,0,0,0,0,32,20,0,0,0,0,0,0,0,0,26,15,0,0,0,0,0,0,15,15,0,0,0,0,0,0,0,0,9,32,9,0,0,0,0,0,32,32,0,23,0,0,0,0,23,0,32,32,0,0,0,0,0,9,32,9] >;

D20⋊C8 in GAP, Magma, Sage, TeX

D_{20}\rtimes C_8
% in TeX

G:=Group("D20:C8");
// GroupNames label

G:=SmallGroup(320,209);
// by ID

G=gap.SmallGroup(320,209);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,100,1123,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^20=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^17*b>;
// generators/relations

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