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G = Q8⋊Dic5order 160 = 25·5

1st semidirect product of Q8 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.9D4, Q81Dic5, C10.5Q16, C10.8SD16, (C5×Q8)⋊4C4, (C2×Q8).1D5, C20.29(C2×C4), C54(Q8⋊C4), (C2×C10).34D4, (C2×C4).40D10, C2.3(Q8⋊D5), (Q8×C10).1C2, C4.2(C2×Dic5), C4.14(C5⋊D4), C4⋊Dic5.10C2, C2.3(C5⋊Q16), (C2×C20).18C22, C2.6(C23.D5), C10.27(C22⋊C4), C22.18(C5⋊D4), (C2×C52C8).5C2, SmallGroup(160,42)

Series: Derived Chief Lower central Upper central

C1C20 — Q8⋊Dic5
C1C5C10C2×C10C2×C20C4⋊Dic5 — Q8⋊Dic5
C5C10C20 — Q8⋊Dic5
C1C22C2×C4C2×Q8

Generators and relations for Q8⋊Dic5
 G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >

2C4
2C4
20C4
2Q8
2C2×C4
10C2×C4
10C8
2C20
2C20
4Dic5
5C4⋊C4
5C2×C8
2C5×Q8
2C52C8
2C2×Dic5
2C2×C20
5Q8⋊C4

Smallest permutation representation of Q8⋊Dic5
Regular action on 160 points
Generators in S160
(1 48 39 12)(2 49 40 13)(3 50 31 14)(4 41 32 15)(5 42 33 16)(6 43 34 17)(7 44 35 18)(8 45 36 19)(9 46 37 20)(10 47 38 11)(21 156 147 138)(22 157 148 139)(23 158 149 140)(24 159 150 131)(25 160 141 132)(26 151 142 133)(27 152 143 134)(28 153 144 135)(29 154 145 136)(30 155 146 137)(51 67 78 84)(52 68 79 85)(53 69 80 86)(54 70 71 87)(55 61 72 88)(56 62 73 89)(57 63 74 90)(58 64 75 81)(59 65 76 82)(60 66 77 83)(91 110 119 125)(92 101 120 126)(93 102 111 127)(94 103 112 128)(95 104 113 129)(96 105 114 130)(97 106 115 121)(98 107 116 122)(99 108 117 123)(100 109 118 124)
(1 74 39 57)(2 75 40 58)(3 76 31 59)(4 77 32 60)(5 78 33 51)(6 79 34 52)(7 80 35 53)(8 71 36 54)(9 72 37 55)(10 73 38 56)(11 89 47 62)(12 90 48 63)(13 81 49 64)(14 82 50 65)(15 83 41 66)(16 84 42 67)(17 85 43 68)(18 86 44 69)(19 87 45 70)(20 88 46 61)(21 111 147 93)(22 112 148 94)(23 113 149 95)(24 114 150 96)(25 115 141 97)(26 116 142 98)(27 117 143 99)(28 118 144 100)(29 119 145 91)(30 120 146 92)(101 137 126 155)(102 138 127 156)(103 139 128 157)(104 140 129 158)(105 131 130 159)(106 132 121 160)(107 133 122 151)(108 134 123 152)(109 135 124 153)(110 136 125 154)
(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 98 6 93)(2 97 7 92)(3 96 8 91)(4 95 9 100)(5 94 10 99)(11 108 16 103)(12 107 17 102)(13 106 18 101)(14 105 19 110)(15 104 20 109)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

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

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

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

Q8⋊Dic5 is a maximal subgroup of
Q8⋊Dic10  Dic5.3Q16  Dic5.9Q16  Q8⋊C4⋊D5  Q8.Dic10  C408C4.C2  Q8.2Dic10  Q8⋊Dic5⋊C2  D5×Q8⋊C4  (Q8×D5)⋊C4  Q8⋊(C4×D5)  Q82D5⋊C4  D10.11SD16  D10.7Q16  (C2×C8).D10  D101C8.C2  C20.48SD16  C20.23Q16  Q8.3Dic10  C4×Q8⋊D5  C42.56D10  C4×C5⋊Q16  C42.59D10  C22⋊Q8.D5  (C2×C10).Q16  C10.(C4○D8)  C52C824D4  C22⋊Q8⋊D5  (C2×C10)⋊Q16  C5⋊(C8.D4)  C42.61D10  C42.62D10  C42.213D10  D20.23D4  C20.Q16  C42.77D10  C205SD16  C20⋊Q16  SD16×Dic5  Dic53SD16  SD16⋊Dic5  (C5×D4).D4  D108SD16  C4014D4  D207D4  C408D4  Dic53Q16  Q16×Dic5  Q16⋊Dic5  (C2×Q16)⋊D5  D105Q16  D20.17D4  D103Q16  C40.36D4  (Q8×C10)⋊16C4  (C5×Q8)⋊13D4  (C2×C10)⋊8Q16  C4○D4⋊Dic5  C20.(C2×D4)  (C5×D4)⋊14D4  (C5×D4).32D4  Dic6⋊Dic5  C10.Dic12  Q82Dic15  Q8⋊Dic15
Q8⋊Dic5 is a maximal quotient of
C20.31C42  C20.26Q16  C10.29C4≀C2  C20.5Q16  C20.10D8  Dic6⋊Dic5  C10.Dic12  Q82Dic15

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20L
order122244444455888810···1020···20
size11112244202022101010102···24···4

34 irreducible representations

dim1111122222222244
type+++++++-+-+-
imageC1C2C2C2C4D4D4D5SD16Q16D10Dic5C5⋊D4C5⋊D4Q8⋊D5C5⋊Q16
kernelQ8⋊Dic5C2×C52C8C4⋊Dic5Q8×C10C5×Q8C20C2×C10C2×Q8C10C10C2×C4Q8C4C22C2C2
# reps1111411222244422

Matrix representation of Q8⋊Dic5 in GL4(𝔽41) generated by

1000
0100
00139
00140
,
1000
0100
002113
00720
,
14000
36600
00400
00040
,
151400
192600
00334
00358
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,21,7,0,0,13,20],[1,36,0,0,40,6,0,0,0,0,40,0,0,0,0,40],[15,19,0,0,14,26,0,0,0,0,33,35,0,0,4,8] >;

Q8⋊Dic5 in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm Dic}_5
% in TeX

G:=Group("Q8:Dic5");
// GroupNames label

G:=SmallGroup(160,42);
// by ID

G=gap.SmallGroup(160,42);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,579,297,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of Q8⋊Dic5 in TeX

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