Copied to
clipboard

## G = Q8×D20order 320 = 26·5

### Direct product of Q8 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Q8×D20
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — Q8×D20
 Lower central C5 — C2×C10 — Q8×D20
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q8×D20
G = < a,b,c,d | a4=c20=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 982 in 280 conjugacy classes, 123 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4×Q8, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, Q8×D5, Q8×C10, C202Q8, C4×D20, D102Q8, Q8×C20, C2×Q8×D5, Q8×D20
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C24, D10, C22×D4, C22×Q8, 2- 1+4, D20, C22×D5, D4×Q8, C2×D20, Q8×D5, C23×D5, C22×D20, C2×Q8×D5, D4.10D10, Q8×D20

Smallest permutation representation of Q8×D20
On 160 points
Generators in S160
(1 120 52 83)(2 101 53 84)(3 102 54 85)(4 103 55 86)(5 104 56 87)(6 105 57 88)(7 106 58 89)(8 107 59 90)(9 108 60 91)(10 109 41 92)(11 110 42 93)(12 111 43 94)(13 112 44 95)(14 113 45 96)(15 114 46 97)(16 115 47 98)(17 116 48 99)(18 117 49 100)(19 118 50 81)(20 119 51 82)(21 72 131 145)(22 73 132 146)(23 74 133 147)(24 75 134 148)(25 76 135 149)(26 77 136 150)(27 78 137 151)(28 79 138 152)(29 80 139 153)(30 61 140 154)(31 62 121 155)(32 63 122 156)(33 64 123 157)(34 65 124 158)(35 66 125 159)(36 67 126 160)(37 68 127 141)(38 69 128 142)(39 70 129 143)(40 71 130 144)
(1 153 52 80)(2 154 53 61)(3 155 54 62)(4 156 55 63)(5 157 56 64)(6 158 57 65)(7 159 58 66)(8 160 59 67)(9 141 60 68)(10 142 41 69)(11 143 42 70)(12 144 43 71)(13 145 44 72)(14 146 45 73)(15 147 46 74)(16 148 47 75)(17 149 48 76)(18 150 49 77)(19 151 50 78)(20 152 51 79)(21 112 131 95)(22 113 132 96)(23 114 133 97)(24 115 134 98)(25 116 135 99)(26 117 136 100)(27 118 137 81)(28 119 138 82)(29 120 139 83)(30 101 140 84)(31 102 121 85)(32 103 122 86)(33 104 123 87)(34 105 124 88)(35 106 125 89)(36 107 126 90)(37 108 127 91)(38 109 128 92)(39 110 129 93)(40 111 130 94)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 31)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)(41 57)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(58 60)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)(121 131)(122 130)(123 129)(124 128)(125 127)(132 140)(133 139)(134 138)(135 137)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 4I 4J 4K 4L ··· 4Q 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 10 10 2 ··· 2 4 4 4 20 ··· 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + + + + - - - image C1 C2 C2 C2 C2 C2 Q8 D4 D5 D10 D10 D10 D20 2- 1+4 Q8×D5 D4.10D10 kernel Q8×D20 C20⋊2Q8 C4×D20 D10⋊2Q8 Q8×C20 C2×Q8×D5 D20 C5×Q8 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C10 C4 C2 # reps 1 3 3 6 1 2 4 4 2 6 6 2 16 1 4 4

Matrix representation of Q8×D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 30 40 0 0 0 0 40 11
,
 37 5 0 0 0 0 13 4 0 0 0 0 0 0 6 40 0 0 0 0 36 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 23 1 0 0 0 0 0 0 40 40 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,40,0,0,0,0,40,11],[37,13,0,0,0,0,5,4,0,0,0,0,0,0,6,36,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Q8×D20 in GAP, Magma, Sage, TeX

Q_8\times D_{20}
% in TeX

G:=Group("Q8xD20");
// GroupNames label

G:=SmallGroup(320,1247);
// by ID

G=gap.SmallGroup(320,1247);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,80,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^20=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽