Copied to
clipboard

## G = D10⋊8SD16order 320 = 26·5

### 2nd semidirect product of D10 and SD16 acting via SD16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D10⋊8SD16
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×Q8×D5 — D10⋊8SD16
 Lower central C5 — C10 — C2×C20 — D10⋊8SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for D108SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=c3 >

Subgroups: 654 in 158 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C2×SD16, C22×Q8, C52C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, Q8⋊D4, C2×C52C8, C4⋊Dic5, D4.D5, C23.D5, C2×C40, C5×SD16, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, C2×C5⋊D4, D4×C10, Q8×C10, C20.44D4, D101C8, Q8⋊Dic5, C2×D4.D5, C202D4, C10×SD16, C2×Q8×D5, D108SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8.C22, C5⋊D4, C22×D5, Q8⋊D4, D4×D5, C2×C5⋊D4, D5×SD16, SD16⋊D5, C23⋊D10, D108SD16

Smallest permutation representation of D108SD16
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 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 50)(20 49)(21 145)(22 144)(23 143)(24 142)(25 141)(26 150)(27 149)(28 148)(29 147)(30 146)(51 74)(52 73)(53 72)(54 71)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 90)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 120)(101 124)(102 123)(103 122)(104 121)(105 130)(106 129)(107 128)(108 127)(109 126)(110 125)(131 154)(132 153)(133 152)(134 151)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)
(1 26 50 153 33 146 20 138)(2 27 41 154 34 147 11 139)(3 28 42 155 35 148 12 140)(4 29 43 156 36 149 13 131)(5 30 44 157 37 150 14 132)(6 21 45 158 38 141 15 133)(7 22 46 159 39 142 16 134)(8 23 47 160 40 143 17 135)(9 24 48 151 31 144 18 136)(10 25 49 152 32 145 19 137)(51 109 69 116 76 123 83 91)(52 110 70 117 77 124 84 92)(53 101 61 118 78 125 85 93)(54 102 62 119 79 126 86 94)(55 103 63 120 80 127 87 95)(56 104 64 111 71 128 88 96)(57 105 65 112 72 129 89 97)(58 106 66 113 73 130 90 98)(59 107 67 114 74 121 81 99)(60 108 68 115 75 122 82 100)
(1 58)(2 59)(3 60)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 67)(12 68)(13 69)(14 70)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 118)(22 119)(23 120)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(91 149)(92 150)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(101 158)(102 159)(103 160)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(121 139)(122 140)(123 131)(124 132)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 8 10 10 2 2 4 4 20 20 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 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 D4 D4 D5 SD16 D10 D10 D10 C5⋊D4 C8.C22 D4×D5 D4×D5 D5×SD16 SD16⋊D5 kernel D10⋊8SD16 C20.44D4 D10⋊1C8 Q8⋊Dic5 C2×D4.D5 C20⋊2D4 C10×SD16 C2×Q8×D5 Dic10 C2×Dic5 C5×Q8 C22×D5 C2×SD16 D10 C2×C8 C2×D4 C2×Q8 Q8 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 4 2 2 2 8 1 2 2 4 4

Matrix representation of D108SD16 in GL4(𝔽41) generated by

 6 6 0 0 35 1 0 0 0 0 1 0 0 0 0 1
,
 6 6 0 0 1 35 0 0 0 0 40 0 0 0 0 40
,
 23 6 0 0 35 18 0 0 0 0 26 15 0 0 26 26
,
 23 6 0 0 35 18 0 0 0 0 19 38 0 0 38 22
G:=sub<GL(4,GF(41))| [6,35,0,0,6,1,0,0,0,0,1,0,0,0,0,1],[6,1,0,0,6,35,0,0,0,0,40,0,0,0,0,40],[23,35,0,0,6,18,0,0,0,0,26,26,0,0,15,26],[23,35,0,0,6,18,0,0,0,0,19,38,0,0,38,22] >;

D108SD16 in GAP, Magma, Sage, TeX

D_{10}\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("D10:8SD16");
// GroupNames label

G:=SmallGroup(320,797);
// by ID

G=gap.SmallGroup(320,797);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,851,438,102,12550]);
// Polycyclic

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

׿
×
𝔽