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G = D10.12D8order 320 = 26·5

1st non-split extension by D10 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.12D8, C2.9(D5×D8), C405C48C2, (C2×C8).9D10, D4⋊C45D5, C10.23(C2×D8), D101C85C2, C4⋊C4.135D10, (C2×D4).23D10, C10.D85C2, D4⋊Dic57C2, C202D4.2C2, (C2×C40).9C22, C4.51(C4○D20), (C2×Dic5).27D4, C22.172(D4×D5), C52(C22.D8), C20.149(C4○D4), C4.78(D42D5), (C2×C20).214C23, (C22×D5).108D4, (D4×C10).35C22, C4⋊Dic5.69C22, C2.11(SD16⋊D5), C10.29(C8.C22), C2.12(D10.12D4), C10.20(C22.D4), (D5×C4⋊C4)⋊3C2, (C5×D4⋊C4)⋊5C2, (C2×C4×D5).12C22, (C2×C10).227(C2×D4), (C5×C4⋊C4).16C22, (C2×C52C8).15C22, (C2×C4).321(C22×D5), SmallGroup(320,401)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.12D8
C1C5C10C2×C10C2×C20C2×C4×D5D5×C4⋊C4 — D10.12D8
C5C10C2×C20 — D10.12D8
C1C22C2×C4D4⋊C4

Generators and relations for D10.12D8
 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=a5c-1 >

Subgroups: 494 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C10, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C22×C4 [×2], C2×D4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, D4⋊C4, C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C52C8, C40, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C22.D8, C2×C52C8, C10.D4, C4⋊Dic5 [×2], C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×C5⋊D4, D4×C10, C10.D8, C405C4, D101C8, D4⋊Dic5, C5×D4⋊C4, D5×C4⋊C4, C202D4, D10.12D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×D8, C8.C22, C22×D5, C22.D8, C4○D20, D4×D5, D42D5, D10.12D4, D5×D8, SD16⋊D5, D10.12D8

Smallest permutation representation of D10.12D8
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 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 143)(22 142)(23 141)(24 150)(25 149)(26 148)(27 147)(28 146)(29 145)(30 144)(51 72)(52 71)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 90)(69 89)(70 88)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 120)(99 119)(100 118)(101 122)(102 121)(103 130)(104 129)(105 128)(106 127)(107 126)(108 125)(109 124)(110 123)(131 152)(132 151)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 153)
(1 149 19 153 33 21 45 136)(2 150 20 154 34 22 46 137)(3 141 11 155 35 23 47 138)(4 142 12 156 36 24 48 139)(5 143 13 157 37 25 49 140)(6 144 14 158 38 26 50 131)(7 145 15 159 39 27 41 132)(8 146 16 160 40 28 42 133)(9 147 17 151 31 29 43 134)(10 148 18 152 32 30 44 135)(51 130 90 118 78 104 64 91)(52 121 81 119 79 105 65 92)(53 122 82 120 80 106 66 93)(54 123 83 111 71 107 67 94)(55 124 84 112 72 108 68 95)(56 125 85 113 73 109 69 96)(57 126 86 114 74 110 70 97)(58 127 87 115 75 101 61 98)(59 128 88 116 76 102 62 99)(60 129 89 117 77 103 63 100)
(1 91 6 96)(2 92 7 97)(3 93 8 98)(4 94 9 99)(5 95 10 100)(11 106 16 101)(12 107 17 102)(13 108 18 103)(14 109 19 104)(15 110 20 105)(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)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,143)(22,142)(23,141)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(51,72)(52,71)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,90)(69,89)(70,88)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,120)(99,119)(100,118)(101,122)(102,121)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(131,152)(132,151)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153), (1,149,19,153,33,21,45,136)(2,150,20,154,34,22,46,137)(3,141,11,155,35,23,47,138)(4,142,12,156,36,24,48,139)(5,143,13,157,37,25,49,140)(6,144,14,158,38,26,50,131)(7,145,15,159,39,27,41,132)(8,146,16,160,40,28,42,133)(9,147,17,151,31,29,43,134)(10,148,18,152,32,30,44,135)(51,130,90,118,78,104,64,91)(52,121,81,119,79,105,65,92)(53,122,82,120,80,106,66,93)(54,123,83,111,71,107,67,94)(55,124,84,112,72,108,68,95)(56,125,85,113,73,109,69,96)(57,126,86,114,74,110,70,97)(58,127,87,115,75,101,61,98)(59,128,88,116,76,102,62,99)(60,129,89,117,77,103,63,100), (1,91,6,96)(2,92,7,97)(3,93,8,98)(4,94,9,99)(5,95,10,100)(11,106,16,101)(12,107,17,102)(13,108,18,103)(14,109,19,104)(15,110,20,105)(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)(11,42)(12,41)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,143)(22,142)(23,141)(24,150)(25,149)(26,148)(27,147)(28,146)(29,145)(30,144)(51,72)(52,71)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,90)(69,89)(70,88)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,120)(99,119)(100,118)(101,122)(102,121)(103,130)(104,129)(105,128)(106,127)(107,126)(108,125)(109,124)(110,123)(131,152)(132,151)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153), (1,149,19,153,33,21,45,136)(2,150,20,154,34,22,46,137)(3,141,11,155,35,23,47,138)(4,142,12,156,36,24,48,139)(5,143,13,157,37,25,49,140)(6,144,14,158,38,26,50,131)(7,145,15,159,39,27,41,132)(8,146,16,160,40,28,42,133)(9,147,17,151,31,29,43,134)(10,148,18,152,32,30,44,135)(51,130,90,118,78,104,64,91)(52,121,81,119,79,105,65,92)(53,122,82,120,80,106,66,93)(54,123,83,111,71,107,67,94)(55,124,84,112,72,108,68,95)(56,125,85,113,73,109,69,96)(57,126,86,114,74,110,70,97)(58,127,87,115,75,101,61,98)(59,128,88,116,76,102,62,99)(60,129,89,117,77,103,63,100), (1,91,6,96)(2,92,7,97)(3,93,8,98)(4,94,9,99)(5,95,10,100)(11,106,16,101)(12,107,17,102)(13,108,18,103)(14,109,19,104)(15,110,20,105)(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),(11,42),(12,41),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,143),(22,142),(23,141),(24,150),(25,149),(26,148),(27,147),(28,146),(29,145),(30,144),(51,72),(52,71),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,90),(69,89),(70,88),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,120),(99,119),(100,118),(101,122),(102,121),(103,130),(104,129),(105,128),(106,127),(107,126),(108,125),(109,124),(110,123),(131,152),(132,151),(133,160),(134,159),(135,158),(136,157),(137,156),(138,155),(139,154),(140,153)], [(1,149,19,153,33,21,45,136),(2,150,20,154,34,22,46,137),(3,141,11,155,35,23,47,138),(4,142,12,156,36,24,48,139),(5,143,13,157,37,25,49,140),(6,144,14,158,38,26,50,131),(7,145,15,159,39,27,41,132),(8,146,16,160,40,28,42,133),(9,147,17,151,31,29,43,134),(10,148,18,152,32,30,44,135),(51,130,90,118,78,104,64,91),(52,121,81,119,79,105,65,92),(53,122,82,120,80,106,66,93),(54,123,83,111,71,107,67,94),(55,124,84,112,72,108,68,95),(56,125,85,113,73,109,69,96),(57,126,86,114,74,110,70,97),(58,127,87,115,75,101,61,98),(59,128,88,116,76,102,62,99),(60,129,89,117,77,103,63,100)], [(1,91,6,96),(2,92,7,97),(3,93,8,98),(4,94,9,99),(5,95,10,100),(11,106,16,101),(12,107,17,102),(13,108,18,103),(14,109,19,104),(15,110,20,105),(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 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444444455888810···1010101010202020202020202040···40
size111181010224420202040224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++++++++--++-
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D8D10D10D10C4○D20C8.C22D42D5D4×D5D5×D8SD16⋊D5
kernelD10.12D8C10.D8C405C4D101C8D4⋊Dic5C5×D4⋊C4D5×C4⋊C4C202D4C2×Dic5C22×D5D4⋊C4C20D10C4⋊C4C2×C8C2×D4C4C10C4C22C2C2
# reps1111111111244222812244

Matrix representation of D10.12D8 in GL6(𝔽41)

4000000
0400000
0003500
0073400
000010
000001
,
4000000
4010000
0073500
0083400
0000400
0000040
,
32180000
3290000
0040000
0004000
0000270
0000038
,
900000
9320000
001000
000100
0000027
0000380

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,35,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,32,0,0,0,0,18,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,27,0,0,0,0,0,0,38],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,38,0,0,0,0,27,0] >;

D10.12D8 in GAP, Magma, Sage, TeX

D_{10}._{12}D_8
% in TeX

G:=Group("D10.12D8");
// GroupNames label

G:=SmallGroup(320,401);
// by ID

G=gap.SmallGroup(320,401);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,851,438,102,12550]);
// 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^-1>;
// generators/relations

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