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## G = Dic5.23D8order 320 = 26·5

### 10th non-split extension by Dic5 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic5.23D8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C20⋊C8 — Dic5.23D8
 Lower central C5 — C10 — C20 — Dic5.23D8
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for Dic5.23D8
G = < a,b,c,d | a10=c8=1, b2=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 322 in 82 conjugacy classes, 32 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic5, Dic5, C20, C2×C10, C2×C10, C4×C8, C4⋊C8, C4×D4, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, D4⋊C8, C4×Dic5, C4⋊Dic5, C23.D5, C2×C5⋊C8, C22×Dic5, D4×C10, C4×C5⋊C8, C20⋊C8, D4×Dic5, Dic5.23D8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), D8, SD16, F5, C22⋊C8, D4⋊C4, C4≀C2, C5⋊C8, C2×F5, D4⋊C8, C2×C5⋊C8, C22.F5, C22⋊F5, D20⋊C4, D4⋊F5, C23.2F5, Dic5.23D8

Smallest permutation representation of Dic5.23D8
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 57 6 52)(2 56 7 51)(3 55 8 60)(4 54 9 59)(5 53 10 58)(11 121 16 126)(12 130 17 125)(13 129 18 124)(14 128 19 123)(15 127 20 122)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)(41 88 46 83)(42 87 47 82)(43 86 48 81)(44 85 49 90)(45 84 50 89)(91 134 96 139)(92 133 97 138)(93 132 98 137)(94 131 99 136)(95 140 100 135)(101 149 106 144)(102 148 107 143)(103 147 108 142)(104 146 109 141)(105 145 110 150)(111 154 116 159)(112 153 117 158)(113 152 118 157)(114 151 119 156)(115 160 120 155)
(1 13 77 110 23 156 89 94)(2 20 76 103 24 153 88 97)(3 17 75 106 25 160 87 100)(4 14 74 109 26 157 86 93)(5 11 73 102 27 154 85 96)(6 18 72 105 28 151 84 99)(7 15 71 108 29 158 83 92)(8 12 80 101 30 155 82 95)(9 19 79 104 21 152 81 98)(10 16 78 107 22 159 90 91)(31 146 69 118 43 137 59 123)(32 143 68 111 44 134 58 126)(33 150 67 114 45 131 57 129)(34 147 66 117 46 138 56 122)(35 144 65 120 47 135 55 125)(36 141 64 113 48 132 54 128)(37 148 63 116 49 139 53 121)(38 145 62 119 50 136 52 124)(39 142 61 112 41 133 51 127)(40 149 70 115 42 140 60 130)
(1 131 57 99 6 136 52 94)(2 138 56 92 7 133 51 97)(3 135 55 95 8 140 60 100)(4 132 54 98 9 137 59 93)(5 139 53 91 10 134 58 96)(11 90 121 44 16 85 126 49)(12 87 130 47 17 82 125 42)(13 84 129 50 18 89 124 45)(14 81 128 43 19 86 123 48)(15 88 127 46 20 83 122 41)(21 146 69 109 26 141 64 104)(22 143 68 102 27 148 63 107)(23 150 67 105 28 145 62 110)(24 147 66 108 29 142 61 103)(25 144 65 101 30 149 70 106)(31 152 74 118 36 157 79 113)(32 159 73 111 37 154 78 116)(33 156 72 114 38 151 77 119)(34 153 71 117 39 158 76 112)(35 160 80 120 40 155 75 115)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + - - + + - image C1 C2 C2 C2 C4 C4 C8 D4 D8 SD16 M4(2) C4≀C2 F5 C2×F5 C5⋊C8 C22.F5 C22⋊F5 D20⋊C4 D4⋊F5 kernel Dic5.23D8 C4×C5⋊C8 C20⋊C8 D4×Dic5 C4⋊Dic5 D4×C10 C5×D4 C2×Dic5 Dic5 Dic5 C20 C10 C2×D4 C2×C4 D4 C4 C22 C2 C2 # reps 1 1 1 1 2 2 8 2 2 2 2 4 1 1 2 2 2 1 1

Matrix representation of Dic5.23D8 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 1 0 0 0 0 0 0 5 35 0 0 0 0 0 0 27 25 40 34 0 0 0 0 32 14 7 7
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 18 21 0 0 0 0 0 0 10 23 0 0 0 0 0 0 6 0 21 26 0 0 0 0 32 33 2 20
,
 15 26 0 0 0 0 0 0 15 15 0 0 0 0 0 0 0 0 4 37 0 0 0 0 0 0 12 37 0 0 0 0 0 0 0 0 0 0 40 1 0 0 0 0 11 25 39 34 0 0 0 0 32 20 16 0 0 0 0 0 14 40 16 0
,
 15 26 0 0 0 0 0 0 26 26 0 0 0 0 0 0 0 0 4 37 0 0 0 0 0 0 37 37 0 0 0 0 0 0 0 0 0 0 40 1 0 0 0 0 11 25 39 34 0 0 0 0 20 5 16 0 0 0 0 0 38 26 16 0

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,5,27,32,0,0,0,0,1,35,25,14,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,18,10,6,32,0,0,0,0,21,23,0,33,0,0,0,0,0,0,21,2,0,0,0,0,0,0,26,20],[15,15,0,0,0,0,0,0,26,15,0,0,0,0,0,0,0,0,4,12,0,0,0,0,0,0,37,37,0,0,0,0,0,0,0,0,0,11,32,14,0,0,0,0,0,25,20,40,0,0,0,0,40,39,16,16,0,0,0,0,1,34,0,0],[15,26,0,0,0,0,0,0,26,26,0,0,0,0,0,0,0,0,4,37,0,0,0,0,0,0,37,37,0,0,0,0,0,0,0,0,0,11,20,38,0,0,0,0,0,25,5,26,0,0,0,0,40,39,16,16,0,0,0,0,1,34,0,0] >;

Dic5.23D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{23}D_8
% in TeX

G:=Group("Dic5.23D8");
// GroupNames label

G:=SmallGroup(320,262);
// by ID

G=gap.SmallGroup(320,262);
# by ID

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

G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

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