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

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

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

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

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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