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## G = C8.2D20order 320 = 26·5

### 2nd non-split extension by C8 of D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C8.2D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×C8⋊D5 — C8.2D20
 Lower central C5 — C10 — C2×C20 — C8.2D20
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C8.2D20
G = < a,b,c | a8=b20=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 454 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C52C8, C40, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C8.D4, C8⋊D5, Dic20, C2×C52C8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C10.Q16, C5×C4.Q8, D102Q8, C2×C8⋊D5, C2×Dic20, C8.2D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8.C22, D20, C22×D5, C8.D4, C2×D20, D4×D5, Q82D5, C4⋊D20, SD16⋊D5, C8.2D20

Smallest permutation representation of C8.2D20
On 160 points
Generators in S160
(1 141 127 91 50 30 117 79)(2 92 118 142 51 80 128 31)(3 143 129 93 52 32 119 61)(4 94 120 144 53 62 130 33)(5 145 131 95 54 34 101 63)(6 96 102 146 55 64 132 35)(7 147 133 97 56 36 103 65)(8 98 104 148 57 66 134 37)(9 149 135 99 58 38 105 67)(10 100 106 150 59 68 136 39)(11 151 137 81 60 40 107 69)(12 82 108 152 41 70 138 21)(13 153 139 83 42 22 109 71)(14 84 110 154 43 72 140 23)(15 155 121 85 44 24 111 73)(16 86 112 156 45 74 122 25)(17 157 123 87 46 26 113 75)(18 88 114 158 47 76 124 27)(19 159 125 89 48 28 115 77)(20 90 116 160 49 78 126 29)
(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 49 50 20)(2 19 51 48)(3 47 52 18)(4 17 53 46)(5 45 54 16)(6 15 55 44)(7 43 56 14)(8 13 57 42)(9 41 58 12)(10 11 59 60)(21 149 152 38)(22 37 153 148)(23 147 154 36)(24 35 155 146)(25 145 156 34)(26 33 157 144)(27 143 158 32)(28 31 159 142)(29 141 160 30)(39 151 150 40)(61 76 93 88)(62 87 94 75)(63 74 95 86)(64 85 96 73)(65 72 97 84)(66 83 98 71)(67 70 99 82)(68 81 100 69)(77 80 89 92)(78 91 90 79)(101 122 131 112)(102 111 132 121)(103 140 133 110)(104 109 134 139)(105 138 135 108)(106 107 136 137)(113 130 123 120)(114 119 124 129)(115 128 125 118)(116 117 126 127)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 20 2 2 8 8 20 40 40 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D20 C8.C22 Q8⋊2D5 D4×D5 SD16⋊D5 kernel C8.2D20 C10.Q16 C5×C4.Q8 D10⋊2Q8 C2×C8⋊D5 C2×Dic20 C40 C2×Dic5 C22×D5 C4.Q8 C20 C4⋊C4 C2×C8 C8 C10 C4 C22 C2 # reps 1 2 1 2 1 1 2 1 1 2 2 4 2 8 2 2 2 8

Matrix representation of C8.2D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 15 32 26 0 0 26 32 15 9 0 0 9 15 9 15 0 0 26 32 26 32
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 2 25 13 19 0 0 16 13 22 23 0 0 13 19 39 16 0 0 22 23 25 28
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 16 39 22 28 0 0 28 25 18 19 0 0 22 28 25 2 0 0 18 19 13 16

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,26,9,26,0,0,15,32,15,32,0,0,32,15,9,26,0,0,26,9,15,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,2,16,13,22,0,0,25,13,19,23,0,0,13,22,39,25,0,0,19,23,16,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,28,22,18,0,0,39,25,28,19,0,0,22,18,25,13,0,0,28,19,2,16] >;

C8.2D20 in GAP, Magma, Sage, TeX

C_8._2D_{20}
% in TeX

G:=Group("C8.2D20");
// GroupNames label

G:=SmallGroup(320,495);
// by ID

G=gap.SmallGroup(320,495);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,555,226,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations

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