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G = D20.3C8order 320 = 26·5

1st non-split extension by D20 of C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.3C8, C40.66D4, C8.31D20, Dic10.3C8, (C2×C80)⋊2C2, (C2×C16)⋊2D5, C4.8(C8×D5), C53(D4.C8), C20.52(C2×C8), C4○D20.5C4, C20.4C89C2, (C2×C8).322D10, C8.45(C5⋊D4), C4.Dic5.5C4, C2.9(D101C8), C10.23(C22⋊C8), (C2×C40).402C22, D20.3C4.3C2, C22.2(C8⋊D5), (C2×C10).27M4(2), C4.41(D10⋊C4), C20.103(C22⋊C4), (C2×C4).97(C4×D5), (C2×C20).404(C2×C4), SmallGroup(320,66)

Series: Derived Chief Lower central Upper central

C1C20 — D20.3C8
C1C5C10C20C40C2×C40D20.3C4 — D20.3C8
C5C10C20 — D20.3C8
C1C8C2×C8C2×C16

Generators and relations for D20.3C8
 G = < a,b,c | a20=b2=1, c8=a10, bab=a-1, ac=ca, cbc-1=a15b >

2C2
20C2
10C22
10C4
2C10
4D5
5D4
5Q8
10C8
10D4
10C2×C4
2Dic5
2D10
2C16
5M4(2)
5C4○D4
10C16
10M4(2)
10C2×C8
2C5⋊D4
2C4×D5
2C52C8
5M5(2)
5C8○D4
2C80
2C8⋊D5
2C8×D5
2C52C16
5D4.C8

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

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

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

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

92 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B8A8B8C8D8E8F8G8H10A···10F16A···16H16I16J16K16L20A···20H40A···40P80A···80AF
order12224444558888888810···1016···161616161620···2040···4080···80
size11220112202211112220202···22···2202020202···22···22···2

92 irreducible representations

dim1111111122222222222
type++++++++
imageC1C2C2C2C4C4C8C8D4D5M4(2)D10D20C5⋊D4C4×D5D4.C8C8×D5C8⋊D5D20.3C8
kernelD20.3C8C20.4C8C2×C80D20.3C4C4.Dic5C4○D20Dic10D20C40C2×C16C2×C10C2×C8C8C8C2×C4C5C4C22C1
# reps11112244222244488832

Matrix representation of D20.3C8 in GL2(𝔽241) generated by

156200
41119
,
0240
2400
,
16171
17084
G:=sub<GL(2,GF(241))| [156,41,200,119],[0,240,240,0],[161,170,71,84] >;

D20.3C8 in GAP, Magma, Sage, TeX

D_{20}._3C_8
% in TeX

G:=Group("D20.3C8");
// GroupNames label

G:=SmallGroup(320,66);
// by ID

G=gap.SmallGroup(320,66);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,100,1123,136,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^2=1,c^8=a^10,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations

Export

Subgroup lattice of D20.3C8 in TeX

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