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G = Dic5⋊SD16order 320 = 26·5

1st semidirect product of Dic5 and SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.11D4, Dic52SD16, C4.96(D4×D5), C4⋊C4.156D10, Q8⋊C414D5, C4.7(C4○D20), C20.128(C2×D4), (C2×C8).126D10, C51(D4.D4), (C2×Q8).23D10, C2.19(D5×SD16), Dic5⋊Q82C2, D208C4.3C2, C20.23(C4○D4), C10.Q1613C2, C20.8Q814C2, (C2×Dic5).43D4, C10.33(C2×SD16), C22.207(D4×D5), C10.27(C4⋊D4), (C2×C40).137C22, (C2×C20).258C23, (C2×D20).72C22, (Q8×C10).41C22, C2.30(D10⋊D4), C2.19(Q16⋊D5), C10.66(C8.C22), (C4×Dic5).31C22, (C2×Dic10).79C22, (C2×Q8⋊D5).3C2, (C2×C40⋊C2).4C2, (C5×Q8⋊C4)⋊14C2, (C2×C10).271(C2×D4), (C5×C4⋊C4).59C22, (C2×C52C8).48C22, (C2×C4).365(C22×D5), SmallGroup(320,445)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic5⋊SD16
C1C5C10C2×C10C2×C20C2×D20D208C4 — Dic5⋊SD16
C5C10C2×C20 — Dic5⋊SD16
C1C22C2×C4Q8⋊C4

Generators and relations for Dic5⋊SD16
 G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=dad=a-1, cbc-1=a5b, bd=db, dcd=c3 >

Subgroups: 534 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×3], Q8 [×4], C23, D5 [×2], C10 [×3], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C52C8, C40, Dic10 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, D4.D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, C10.D4 [×2], D10⋊C4, Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, Q8×C10, C10.Q16, C20.8Q8, C5×Q8⋊C4, D208C4, C2×C40⋊C2, C2×Q8⋊D5, Dic5⋊Q8, Dic5⋊SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8.C22, C22×D5, D4.D4, C4○D20, D4×D5 [×2], D10⋊D4, D5×SD16, Q16⋊D5, Dic5⋊SD16

Smallest permutation representation of Dic5⋊SD16
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 93 6 98)(2 92 7 97)(3 91 8 96)(4 100 9 95)(5 99 10 94)(11 127 16 122)(12 126 17 121)(13 125 18 130)(14 124 19 129)(15 123 20 128)(21 77 26 72)(22 76 27 71)(23 75 28 80)(24 74 29 79)(25 73 30 78)(31 112 36 117)(32 111 37 116)(33 120 38 115)(34 119 39 114)(35 118 40 113)(41 102 46 107)(42 101 47 106)(43 110 48 105)(44 109 49 104)(45 108 50 103)(51 147 56 142)(52 146 57 141)(53 145 58 150)(54 144 59 149)(55 143 60 148)(61 137 66 132)(62 136 67 131)(63 135 68 140)(64 134 69 139)(65 133 70 138)(81 156 86 151)(82 155 87 160)(83 154 88 159)(84 153 89 158)(85 152 90 157)
(1 83 20 79 39 65 43 58)(2 82 11 78 40 64 44 57)(3 81 12 77 31 63 45 56)(4 90 13 76 32 62 46 55)(5 89 14 75 33 61 47 54)(6 88 15 74 34 70 48 53)(7 87 16 73 35 69 49 52)(8 86 17 72 36 68 50 51)(9 85 18 71 37 67 41 60)(10 84 19 80 38 66 42 59)(21 112 140 108 147 91 151 126)(22 111 131 107 148 100 152 125)(23 120 132 106 149 99 153 124)(24 119 133 105 150 98 154 123)(25 118 134 104 141 97 155 122)(26 117 135 103 142 96 156 121)(27 116 136 102 143 95 157 130)(28 115 137 101 144 94 158 129)(29 114 138 110 145 93 159 128)(30 113 139 109 146 92 160 127)
(2 10)(3 9)(4 8)(5 7)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 157)(22 156)(23 155)(24 154)(25 153)(26 152)(27 151)(28 160)(29 159)(30 158)(31 37)(32 36)(33 35)(38 40)(51 62)(52 61)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(71 81)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 95)(92 94)(96 100)(97 99)(101 127)(102 126)(103 125)(104 124)(105 123)(106 122)(107 121)(108 130)(109 129)(110 128)(111 117)(112 116)(113 115)(118 120)(131 142)(132 141)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111120202244810102040224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type++++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D4D5SD16C4○D4D10D10D10C4○D20C8.C22D4×D5D4×D5D5×SD16Q16⋊D5
kernelDic5⋊SD16C10.Q16C20.8Q8C5×Q8⋊C4D208C4C2×C40⋊C2C2×Q8⋊D5Dic5⋊Q8D20C2×Dic5Q8⋊C4Dic5C20C4⋊C4C2×C8C2×Q8C4C10C4C22C2C2
# reps1111111122242222812244

Matrix representation of Dic5⋊SD16 in GL4(𝔽41) generated by

1000
0100
003540
0010
,
40000
04000
001328
003228
,
01700
293000
002121
001820
,
11800
04000
00635
004035
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,35,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,13,32,0,0,28,28],[0,29,0,0,17,30,0,0,0,0,21,18,0,0,21,20],[1,0,0,0,18,40,0,0,0,0,6,40,0,0,35,35] >;

Dic5⋊SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes {\rm SD}_{16}
% in TeX

G:=Group("Dic5:SD16");
// GroupNames label

G:=SmallGroup(320,445);
// by ID

G=gap.SmallGroup(320,445);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,1684,851,438,102,12550]);
// Polycyclic

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

׿
×
𝔽