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G = Dic5.14D8order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D41Dic10, Dic5.14D8, C20⋊Q81C2, (C5×D4)⋊1Q8, C2.6(D5×D8), C4⋊C4.1D10, C405C46C2, (C2×C8).6D10, C20.1(C2×Q8), C51(D4⋊Q8), C10.19(C2×D8), D4⋊C4.2D5, C10.D83C2, (C2×C40).6C22, C20.8Q84C2, (D4×Dic5).3C2, C4.1(C2×Dic10), (C2×D4).126D10, D4⋊Dic5.1C2, C22.161(D4×D5), C10.7(C22⋊Q8), C20.145(C4○D4), C4.74(D42D5), (C2×C20).199C23, (C2×Dic5).190D4, C2.8(SD16⋊D5), (D4×C10).20C22, C4⋊Dic5.59C22, C10.25(C8.C22), (C4×Dic5).10C22, C2.12(Dic5.14D4), (C5×C4⋊C4).4C22, (C5×D4⋊C4).2C2, (C2×C10).212(C2×D4), (C2×C52C8).6C22, (C2×C4).306(C22×D5), SmallGroup(320,386)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic5.14D8
C1C5C10C2×C10C2×C20C4×Dic5D4×Dic5 — Dic5.14D8
C5C10C2×C20 — Dic5.14D8
C1C22C2×C4D4⋊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 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4, Q8 [×2], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20, C2×C10, C2×C10 [×4], D4⋊C4, D4⋊C4, C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, D4⋊Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], 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 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, D8 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×D8, C8.C22, Dic10 [×2], 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)(11 129 16 124)(12 128 17 123)(13 127 18 122)(14 126 19 121)(15 125 20 130)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 100 36 95)(32 99 37 94)(33 98 38 93)(34 97 39 92)(35 96 40 91)(41 108 46 103)(42 107 47 102)(43 106 48 101)(44 105 49 110)(45 104 50 109)(51 153 56 158)(52 152 57 157)(53 151 58 156)(54 160 59 155)(55 159 60 154)(71 140 76 135)(72 139 77 134)(73 138 78 133)(74 137 79 132)(75 136 80 131)(81 145 86 150)(82 144 87 149)(83 143 88 148)(84 142 89 147)(85 141 90 146)
(1 53 13 85 39 74 43 64)(2 54 14 86 40 75 44 65)(3 55 15 87 31 76 45 66)(4 56 16 88 32 77 46 67)(5 57 17 89 33 78 47 68)(6 58 18 90 34 79 48 69)(7 59 19 81 35 80 49 70)(8 60 20 82 36 71 50 61)(9 51 11 83 37 72 41 62)(10 52 12 84 38 73 42 63)(21 111 151 122 141 97 137 101)(22 112 152 123 142 98 138 102)(23 113 153 124 143 99 139 103)(24 114 154 125 144 100 140 104)(25 115 155 126 145 91 131 105)(26 116 156 127 146 92 132 106)(27 117 157 128 147 93 133 107)(28 118 158 129 148 94 134 108)(29 119 159 130 149 95 135 109)(30 120 160 121 150 96 136 110)
(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(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)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 121)(131 150)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444455888810···1010101010202020202020202040···40
size111144228101020202040224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++-+++++---++-
imageC1C2C2C2C2C2C2C2D4Q8D5D8C4○D4D10D10D10Dic10C8.C22D42D5D4×D5D5×D8SD16⋊D5
kernelDic5.14D8C10.D8C20.8Q8C405C4D4⋊Dic5C5×D4⋊C4C20⋊Q8D4×Dic5C2×Dic5C5×D4D4⋊C4Dic5C20C4⋊C4C2×C8C2×D4D4C10C4C22C2C2
# reps1111111122242222812244

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

1000
0100
00140
00366
,
40000
04000
002317
00518
,
242400
29000
003932
00372
,
1000
404000
0010
0001
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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