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## G = D10.12SD16order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10.12SD16
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — D5×C4⋊C4 — D10.12SD16
 Lower central C5 — C10 — C2×C20 — D10.12SD16
 Upper central C1 — C22 — C2×C4 — C4.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 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 10 10 2 2 4 4 8 20 20 20 40 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 SD16 D10 D10 C4○D20 C8.C22 Q8⋊2D5 D4×D5 D5×SD16 SD16⋊D5 kernel D10.12SD16 C20.Q8 C10.Q16 C20.44D4 D10⋊1C8 C5×C4.Q8 D5×C4⋊C4 D10⋊2Q8 C2×Dic5 C22×D5 C4.Q8 C20 D10 C4⋊C4 C2×C8 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 4 4 4 2 8 1 2 2 4 4

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

 1 0 0 0 0 1 0 0 0 0 1 34 0 0 7 34
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 7 40
,
 15 26 0 0 15 15 0 0 0 0 17 1 0 0 40 24
,
 17 32 0 0 32 24 0 0 0 0 9 0 0 0 0 9
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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