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G = D10.12SD16order 320 = 26·5

2nd non-split extension by D10 of SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.12SD16, C4.Q86D5, C4⋊C4.36D10, (C2×C8).137D10, C2.22(D5×SD16), C20.27(C4○D4), C4.72(C4○D20), C10.Q1615C2, D102Q8.5C2, C20.Q816C2, (C2×Dic5).48D4, C10.38(C2×SD16), C22.214(D4×D5), D101C8.13C2, C20.44D431C2, (C2×C20).278C23, (C2×C40).284C22, C4.24(Q82D5), (C22×D5).119D4, C53(C23.47D4), C2.23(SD16⋊D5), C10.42(C8.C22), C4⋊Dic5.110C22, (C2×Dic10).86C22, C2.11(D10.13D4), C10.41(C22.D4), (D5×C4⋊C4).6C2, (C5×C4.Q8)⋊15C2, (C2×C4×D5).35C22, (C2×C10).283(C2×D4), (C5×C4⋊C4).71C22, (C2×C52C8).56C22, (C2×C4).381(C22×D5), SmallGroup(320,489)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.12SD16
C1C5C10C2×C10C2×C20C2×C4×D5D5×C4⋊C4 — D10.12SD16
C5C10C2×C20 — D10.12SD16
C1C22C2×C4C4.Q8

Generators and relations for D10.12SD16
 G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c3 >

Subgroups: 430 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×9], Q8 [×2], C23, D5 [×2], C10 [×3], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, C22×C4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C22⋊C8, Q8⋊C4 [×2], C4.Q8, C4.Q8, C2×C4⋊C4, C22⋊Q8, C52C8, C40, Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.47D4, C2×C52C8, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×C4×D5, C20.Q8, C10.Q16, C20.44D4, D101C8, C5×C4.Q8, D5×C4⋊C4, D102Q8, D10.12SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8.C22, C22×D5, C23.47D4, C4○D20, D4×D5, Q82D5, D10.13D4, D5×SD16, SD16⋊D5, D10.12SD16

Smallest permutation representation of D10.12SD16
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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 50)(21 145)(22 144)(23 143)(24 142)(25 141)(26 150)(27 149)(28 148)(29 147)(30 146)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 80)(60 79)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 90)(91 113)(92 112)(93 111)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 124)(102 123)(103 122)(104 121)(105 130)(106 129)(107 128)(108 127)(109 126)(110 125)(131 158)(132 157)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 160)(140 159)
(1 90 50 73 33 66 11 52)(2 81 41 74 34 67 12 53)(3 82 42 75 35 68 13 54)(4 83 43 76 36 69 14 55)(5 84 44 77 37 70 15 56)(6 85 45 78 38 61 16 57)(7 86 46 79 39 62 17 58)(8 87 47 80 40 63 18 59)(9 88 48 71 31 64 19 60)(10 89 49 72 32 65 20 51)(21 118 137 125 141 97 158 101)(22 119 138 126 142 98 159 102)(23 120 139 127 143 99 160 103)(24 111 140 128 144 100 151 104)(25 112 131 129 145 91 152 105)(26 113 132 130 146 92 153 106)(27 114 133 121 147 93 154 107)(28 115 134 122 148 94 155 108)(29 116 135 123 149 95 156 109)(30 117 136 124 150 96 157 110)
(1 97 6 92)(2 98 7 93)(3 99 8 94)(4 100 9 95)(5 91 10 96)(11 101 16 106)(12 102 17 107)(13 103 18 108)(14 104 19 109)(15 105 20 110)(21 85 26 90)(22 86 27 81)(23 87 28 82)(24 88 29 83)(25 89 30 84)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111110102244820202040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4SD16D10D10C4○D20C8.C22Q82D5D4×D5D5×SD16SD16⋊D5
kernelD10.12SD16C20.Q8C10.Q16C20.44D4D101C8C5×C4.Q8D5×C4⋊C4D102Q8C2×Dic5C22×D5C4.Q8C20D10C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111111124442812244

Matrix representation of D10.12SD16 in GL4(𝔽41) generated by

1000
0100
00134
00734
,
40000
04000
0010
00740
,
152600
151500
00171
004024
,
173200
322400
0090
0009
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,7,0,0,34,34],[40,0,0,0,0,40,0,0,0,0,1,7,0,0,0,40],[15,15,0,0,26,15,0,0,0,0,17,40,0,0,1,24],[17,32,0,0,32,24,0,0,0,0,9,0,0,0,0,9] >;

D10.12SD16 in GAP, Magma, Sage, TeX

D_{10}._{12}{\rm SD}_{16}
% in TeX

G:=Group("D10.12SD16");
// GroupNames label

G:=SmallGroup(320,489);
// by ID

G=gap.SmallGroup(320,489);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,100,851,102,12550]);
// Polycyclic

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

׿
×
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