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G = D10.11SD16order 320 = 26·5

1st non-split extension by D10 of SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.11SD16, C406C414C2, C4⋊C4.151D10, Q8⋊C410D5, Q8⋊Dic57C2, (C2×C8).122D10, (C2×Q8).15D10, C2.16(D5×SD16), D101C8.5C2, C4.55(C4○D20), D103Q8.2C2, C20.Q810C2, (C2×Dic5).37D4, C10.30(C2×SD16), C22.195(D4×D5), C20.161(C4○D4), C4.86(D42D5), (C2×C40).133C22, (C2×C20).245C23, (C22×D5).113D4, C52(C23.47D4), C4⋊Dic5.93C22, (Q8×C10).28C22, C2.13(Q16⋊D5), C10.59(C8.C22), C2.16(D10.12D4), C10.24(C22.D4), (D5×C4⋊C4).2C2, (C2×C4×D5).23C22, (C5×Q8⋊C4)⋊10C2, (C2×C10).258(C2×D4), (C5×C4⋊C4).46C22, (C2×C52C8).37C22, (C2×C4).352(C22×D5), SmallGroup(320,432)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.11SD16
C1C5C10C2×C10C2×C20C2×C4×D5D5×C4⋊C4 — D10.11SD16
C5C10C2×C20 — D10.11SD16
C1C22C2×C4Q8⋊C4

Generators and relations for D10.11SD16
 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=a5c3 >

Subgroups: 414 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, C4⋊C4 [×4], 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, Q8⋊C4, C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, C52C8, C40, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C23.47D4, C2×C52C8, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, Q8×C10, C20.Q8, C406C4, D101C8, Q8⋊Dic5, C5×Q8⋊C4, D5×C4⋊C4, D103Q8, D10.11SD16
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, D42D5, D10.12D4, D5×SD16, Q16⋊D5, D10.11SD16

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

dim1111111122222222244444
type++++++++++++++--+
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4SD16D10D10D10C4○D20C8.C22D42D5D4×D5D5×SD16Q16⋊D5
kernelD10.11SD16C20.Q8C406C4D101C8Q8⋊Dic5C5×Q8⋊C4D5×C4⋊C4D103Q8C2×Dic5C22×D5Q8⋊C4C20D10C4⋊C4C2×C8C2×Q8C4C10C4C22C2C2
# reps1111111111244222812244

Matrix representation of D10.11SD16 in GL4(𝔽41) generated by

1000
0100
0016
00356
,
40000
04000
0010
003540
,
262600
152600
00228
001339
,
282300
231300
0090
0009
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,35,0,0,6,6],[40,0,0,0,0,40,0,0,0,0,1,35,0,0,0,40],[26,15,0,0,26,26,0,0,0,0,2,13,0,0,28,39],[28,23,0,0,23,13,0,0,0,0,9,0,0,0,0,9] >;

D10.11SD16 in GAP, Magma, Sage, TeX

D_{10}._{11}{\rm SD}_{16}
% in TeX

G:=Group("D10.11SD16");
// GroupNames label

G:=SmallGroup(320,432);
// by ID

G=gap.SmallGroup(320,432);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,184,851,438,102,12550]);
// 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=a^5*c^3>;
// generators/relations

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