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## G = Dic5.14D8order 320 = 26·5

### 1st non-split extension by Dic5 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5.14D8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — D4×Dic5 — Dic5.14D8
 Lower central C5 — C10 — C2×C20 — Dic5.14D8
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic5.14D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=a5c-1 >

Subgroups: 422 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C52C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4⋊Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C22×Dic5, D4×C10, C10.D8, C20.8Q8, C405C4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, D4×Dic5, Dic5.14D8
Quotients: C1, C2, C22, D4, Q8, C23, D5, D8, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×D8, C8.C22, Dic10, C22×D5, D4⋊Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D5×D8, SD16⋊D5, Dic5.14D8

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

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + + - - - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 D8 C4○D4 D10 D10 D10 Dic10 C8.C22 D4⋊2D5 D4×D5 D5×D8 SD16⋊D5 kernel Dic5.14D8 C10.D8 C20.8Q8 C40⋊5C4 D4⋊Dic5 C5×D4⋊C4 C20⋊Q8 D4×Dic5 C2×Dic5 C5×D4 D4⋊C4 Dic5 C20 C4⋊C4 C2×C8 C2×D4 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 2 4 4

Matrix representation of Dic5.14D8 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 40 0 0 36 6
,
 40 0 0 0 0 40 0 0 0 0 23 17 0 0 5 18
,
 24 24 0 0 29 0 0 0 0 0 39 32 0 0 37 2
,
 1 0 0 0 40 40 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,36,0,0,40,6],[40,0,0,0,0,40,0,0,0,0,23,5,0,0,17,18],[24,29,0,0,24,0,0,0,0,0,39,37,0,0,32,2],[1,40,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;

Dic5.14D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{14}D_8
% in TeX

G:=Group("Dic5.14D8");
// GroupNames label

G:=SmallGroup(320,386);
// by ID

G=gap.SmallGroup(320,386);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,254,219,58,851,438,102,12550]);
// Polycyclic

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

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