Copied to
clipboard

## G = Dic5⋊4D8order 320 = 26·5

### 1st semidirect product of Dic5 and D8 acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic5⋊4D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — D4×Dic5 — Dic5⋊4D8
 Lower central C5 — C10 — C20 — Dic5⋊4D8
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic54D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 566 in 134 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×C8, D4⋊C4, D4⋊C4, C2.D8, C4×D4, C2×D8, C52C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C4×D8, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, D4⋊D5, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C22×Dic5, D4×C10, C10.D8, C8×Dic5, D205C4, C5×D4⋊C4, D208C4, C2×D4⋊D5, D4×Dic5, Dic54D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, D8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×D8, C4○D8, C4×D5, C22×D5, C4×D8, C2×C4×D5, D4×D5, D42D5, Dic54D4, D5×D8, SD163D5, Dic54D8

Smallest permutation representation of Dic54D8
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 68 6 63)(2 67 7 62)(3 66 8 61)(4 65 9 70)(5 64 10 69)(11 126 16 121)(12 125 17 130)(13 124 18 129)(14 123 19 128)(15 122 20 127)(21 51 26 56)(22 60 27 55)(23 59 28 54)(24 58 29 53)(25 57 30 52)(31 77 36 72)(32 76 37 71)(33 75 38 80)(34 74 39 79)(35 73 40 78)(41 85 46 90)(42 84 47 89)(43 83 48 88)(44 82 49 87)(45 81 50 86)(91 141 96 146)(92 150 97 145)(93 149 98 144)(94 148 99 143)(95 147 100 142)(101 131 106 136)(102 140 107 135)(103 139 108 134)(104 138 109 133)(105 137 110 132)(111 157 116 152)(112 156 117 151)(113 155 118 160)(114 154 119 159)(115 153 120 158)
(1 114 34 94 28 128 48 108)(2 113 35 93 29 127 49 107)(3 112 36 92 30 126 50 106)(4 111 37 91 21 125 41 105)(5 120 38 100 22 124 42 104)(6 119 39 99 23 123 43 103)(7 118 40 98 24 122 44 102)(8 117 31 97 25 121 45 101)(9 116 32 96 26 130 46 110)(10 115 33 95 27 129 47 109)(11 81 131 61 151 77 145 57)(12 90 132 70 152 76 146 56)(13 89 133 69 153 75 147 55)(14 88 134 68 154 74 148 54)(15 87 135 67 155 73 149 53)(16 86 136 66 156 72 150 52)(17 85 137 65 157 71 141 51)(18 84 138 64 158 80 142 60)(19 83 139 63 159 79 143 59)(20 82 140 62 160 78 144 58)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D5 D8 C4○D4 D10 D10 D10 C4○D8 C4×D5 D4⋊2D5 D4×D5 D5×D8 SD16⋊3D5 kernel Dic5⋊4D8 C10.D8 C8×Dic5 D20⋊5C4 C5×D4⋊C4 D20⋊8C4 C2×D4⋊D5 D4×Dic5 D4⋊D5 C2×Dic5 D4⋊C4 Dic5 C20 C4⋊C4 C2×C8 C2×D4 C10 D4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 4 2 2 2 2 4 8 2 2 4 4

Matrix representation of Dic54D8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 7 40 0 0 1 0
,
 9 0 0 0 0 9 0 0 0 0 22 19 0 0 9 19
,
 0 12 0 0 17 17 0 0 0 0 7 34 0 0 1 34
,
 0 12 0 0 24 0 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,7,1,0,0,40,0],[9,0,0,0,0,9,0,0,0,0,22,9,0,0,19,19],[0,17,0,0,12,17,0,0,0,0,7,1,0,0,34,34],[0,24,0,0,12,0,0,0,0,0,40,0,0,0,0,40] >;

Dic54D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_4D_8
% in TeX

G:=Group("Dic5:4D8");
// GroupNames label

G:=SmallGroup(320,383);
// by ID

G=gap.SmallGroup(320,383);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,135,100,570,297,136,12550]);
// Polycyclic

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

׿
×
𝔽