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## G = Dic10.C8order 320 = 26·5

### 2nd non-split extension by Dic10 of C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic10.C8
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16 — Dic10.C8
 Lower central C5 — C10 — Dic10.C8
 Upper central C1 — C4 — M4(2)

Generators and relations for Dic10.C8
G = < a,b,c | a20=1, b2=c8=a10, bab-1=a-1, cac-1=a13, bc=cb >

Subgroups: 226 in 84 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D5 [×2], C10, C10, C16 [×4], C2×C8 [×3], M4(2), M4(2) [×2], C4○D4, Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C2×C16 [×3], M5(2) [×3], C8○D4, C52C8 [×2], C40 [×2], Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, D4○C16, C5⋊C16 [×2], C5⋊C16 [×2], C8×D5 [×2], C8⋊D5 [×2], C2×C52C8, C5×M4(2), C4○D20, D5⋊C16 [×2], C8.F5 [×2], C2×C5⋊C16, C20.C8, D20.2C4, Dic10.C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, C22×C8, C2×F5 [×3], D4○C16, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, Dic10.C8

Smallest permutation representation of Dic10.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)
(1 139 11 129)(2 138 12 128)(3 137 13 127)(4 136 14 126)(5 135 15 125)(6 134 16 124)(7 133 17 123)(8 132 18 122)(9 131 19 121)(10 130 20 140)(21 92 31 82)(22 91 32 81)(23 90 33 100)(24 89 34 99)(25 88 35 98)(26 87 36 97)(27 86 37 96)(28 85 38 95)(29 84 39 94)(30 83 40 93)(41 65 51 75)(42 64 52 74)(43 63 53 73)(44 62 54 72)(45 61 55 71)(46 80 56 70)(47 79 57 69)(48 78 58 68)(49 77 59 67)(50 76 60 66)(101 149 111 159)(102 148 112 158)(103 147 113 157)(104 146 114 156)(105 145 115 155)(106 144 116 154)(107 143 117 153)(108 142 118 152)(109 141 119 151)(110 160 120 150)
(1 118 75 33 134 147 56 95 11 108 65 23 124 157 46 85)(2 115 64 26 135 144 45 88 12 105 74 36 125 154 55 98)(3 112 73 39 136 141 54 81 13 102 63 29 126 151 44 91)(4 109 62 32 137 158 43 94 14 119 72 22 127 148 53 84)(5 106 71 25 138 155 52 87 15 116 61 35 128 145 42 97)(6 103 80 38 139 152 41 100 16 113 70 28 129 142 51 90)(7 120 69 31 140 149 50 93 17 110 79 21 130 159 60 83)(8 117 78 24 121 146 59 86 18 107 68 34 131 156 49 96)(9 114 67 37 122 143 48 99 19 104 77 27 132 153 58 89)(10 111 76 30 123 160 57 92 20 101 66 40 133 150 47 82)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 16A ··· 16H 16I ··· 16T 20A 20B 20C 40A 40B 40C 40D order 1 2 2 2 2 4 4 4 4 4 5 8 8 8 8 8 8 8 8 8 8 10 10 16 ··· 16 16 ··· 16 20 20 20 40 40 40 40 size 1 1 2 10 10 1 1 2 10 10 4 2 2 2 2 5 5 5 5 10 10 4 8 5 ··· 5 10 ··· 10 4 4 8 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 4 8 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D4○C16 F5 C2×F5 C2×F5 D5⋊C8 D5⋊C8 Dic10.C8 kernel Dic10.C8 D5⋊C16 C8.F5 C2×C5⋊C16 C20.C8 D20.2C4 C8⋊D5 C5×M4(2) C4○D20 Dic10 D20 C5⋊D4 C5 M4(2) C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 8 8 1 2 1 2 2 2

Matrix representation of Dic10.C8 in GL6(𝔽241)

 0 16 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 1 0 0 240 0 0 1 0 0 0 240 0 1 0 0 0 0 240 1
,
 0 60 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240 0 0 0
,
 115 0 0 0 0 0 0 115 0 0 0 0 0 0 227 14 145 0 0 0 131 14 0 227 0 0 227 0 14 131 0 0 0 145 14 227

G:=sub<GL(6,GF(241))| [0,15,0,0,0,0,16,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,1,1,1],[0,4,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240,0,0,0],[115,0,0,0,0,0,0,115,0,0,0,0,0,0,227,131,227,0,0,0,14,14,0,145,0,0,145,0,14,14,0,0,0,227,131,227] >;

Dic10.C8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}.C_8
% in TeX

G:=Group("Dic10.C8");
// GroupNames label

G:=SmallGroup(320,1063);
// by ID

G=gap.SmallGroup(320,1063);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=1,b^2=c^8=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^13,b*c=c*b>;
// generators/relations

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