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G = Dic57SD16order 320 = 26·5

2nd semidirect product of Dic5 and SD16 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic57SD16, Q8⋊D55C4, Q81(C4×D5), C54(C4×SD16), (Q8×Dic5)⋊1C2, C10.64(C4×D4), C2.3(D5×SD16), C4⋊C4.142D10, Q8⋊C421D5, (C8×Dic5)⋊21C2, D20.16(C2×C4), (C2×C8).204D10, (C2×Q8).96D10, C20.Q89C2, C22.73(D4×D5), D208C4.1C2, D205C4.6C2, C10.66(C4○D8), C20.44(C22×C4), C10.26(C2×SD16), C20.154(C4○D4), C2.1(Q8.D10), C4.51(D42D5), (C2×C40).189C22, (C2×C20).228C23, (C2×Dic5).270D4, (C2×D20).60C22, C4⋊Dic5.78C22, (Q8×C10).11C22, C2.18(Dic54D4), (C4×Dic5).255C22, C4.9(C2×C4×D5), C52C819(C2×C4), (C5×Q8)⋊10(C2×C4), (C2×Q8⋊D5).1C2, (C5×Q8⋊C4)⋊15C2, (C2×C10).241(C2×D4), (C5×C4⋊C4).29C22, (C2×C4).335(C22×D5), (C2×C52C8).222C22, SmallGroup(320,415)

Series: Derived Chief Lower central Upper central

C1C20 — Dic57SD16
C1C5C10C20C2×C20C4×Dic5D208C4 — Dic57SD16
C5C10C20 — Dic57SD16
C1C22C2×C4Q8⋊C4

Generators and relations for Dic57SD16
 G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c3 >

Subgroups: 486 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C5, C8 [×3], C2×C4, C2×C4 [×7], D4 [×3], Q8 [×2], Q8, C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×3], D10 [×4], C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C52C8 [×2], C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C4×SD16, C2×C52C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4, Q8⋊D5 [×4], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, Q8×C10, C20.Q8, C8×Dic5, D205C4, C5×Q8⋊C4, D208C4, C2×Q8⋊D5, Q8×Dic5, Dic57SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, SD16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×SD16, C4○D8, C4×D5 [×2], C22×D5, C4×SD16, C2×C4×D5, D4×D5, D42D5, Dic54D4, D5×SD16, Q8.D10, Dic57SD16

Smallest permutation representation of Dic57SD16
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)
(1 108 6 103)(2 107 7 102)(3 106 8 101)(4 105 9 110)(5 104 10 109)(11 58 16 53)(12 57 17 52)(13 56 18 51)(14 55 19 60)(15 54 20 59)(21 97 26 92)(22 96 27 91)(23 95 28 100)(24 94 29 99)(25 93 30 98)(31 119 36 114)(32 118 37 113)(33 117 38 112)(34 116 39 111)(35 115 40 120)(41 125 46 130)(42 124 47 129)(43 123 48 128)(44 122 49 127)(45 121 50 126)(61 157 66 152)(62 156 67 151)(63 155 68 160)(64 154 69 159)(65 153 70 158)(71 147 76 142)(72 146 77 141)(73 145 78 150)(74 144 79 149)(75 143 80 148)(81 137 86 132)(82 136 87 131)(83 135 88 140)(84 134 89 139)(85 133 90 138)
(1 80 40 60 28 88 48 68)(2 79 31 59 29 87 49 67)(3 78 32 58 30 86 50 66)(4 77 33 57 21 85 41 65)(5 76 34 56 22 84 42 64)(6 75 35 55 23 83 43 63)(7 74 36 54 24 82 44 62)(8 73 37 53 25 81 45 61)(9 72 38 52 26 90 46 70)(10 71 39 51 27 89 47 69)(11 93 137 121 157 101 145 113)(12 92 138 130 158 110 146 112)(13 91 139 129 159 109 147 111)(14 100 140 128 160 108 148 120)(15 99 131 127 151 107 149 119)(16 98 132 126 152 106 150 118)(17 97 133 125 153 105 141 117)(18 96 134 124 154 104 142 116)(19 95 135 123 155 103 143 115)(20 94 136 122 156 102 144 114)
(1 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 30)(10 29)(11 133)(12 132)(13 131)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(31 39)(32 38)(33 37)(34 36)(41 45)(42 44)(46 50)(47 49)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 90)(59 89)(60 88)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 80)(69 79)(70 78)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 110)(99 109)(100 108)(111 119)(112 118)(113 117)(114 116)(121 125)(122 124)(126 130)(127 129)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 160)(149 159)(150 158)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444444444558888888810···102020202020···2040···40
size11112020224444555510102020222222101010102···244448···84···4

56 irreducible representations

dim1111111112222222224444
type+++++++++++++-++
imageC1C2C2C2C2C2C2C2C4D4D5SD16C4○D4D10D10D10C4○D8C4×D5D42D5D4×D5D5×SD16Q8.D10
kernelDic57SD16C20.Q8C8×Dic5D205C4C5×Q8⋊C4D208C4C2×Q8⋊D5Q8×Dic5Q8⋊D5C2×Dic5Q8⋊C4Dic5C20C4⋊C4C2×C8C2×Q8C10Q8C4C22C2C2
# reps1111111182242222482244

Matrix representation of Dic57SD16 in GL4(𝔽41) generated by

04000
1700
0010
0001
,
22900
191900
00400
00040
,
7100
343400
001130
00260
,
344000
7700
00400
00401
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[22,19,0,0,9,19,0,0,0,0,40,0,0,0,0,40],[7,34,0,0,1,34,0,0,0,0,11,26,0,0,30,0],[34,7,0,0,40,7,0,0,0,0,40,40,0,0,0,1] >;

Dic57SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_7{\rm SD}_{16}
% in TeX

G:=Group("Dic5:7SD16");
// GroupNames label

G:=SmallGroup(320,415);
// by ID

G=gap.SmallGroup(320,415);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,120,135,184,570,297,136,12550]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

׿
×
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