Copied to
clipboard

G = D2010Q8order 320 = 26·5

The semidirect product of D20 and Q8 acting through Inn(D20)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2010Q8, C42.130D10, C10.1112+ 1+4, (C4×Q8)⋊12D5, C4.50(Q8×D5), (Q8×C20)⋊14C2, C54(D43Q8), C4⋊C4.326D10, D103Q88C2, C202Q828C2, (C4×D20).21C2, D10.20(C2×Q8), C20.108(C2×Q8), D10⋊Q811C2, C4.67(C4○D20), (C2×Q8).178D10, C20.6Q818C2, C20.118(C4○D4), C10.31(C22×Q8), (C2×C10).123C24, (C4×C20).175C22, (C2×C20).590C23, C2.23(D48D10), (C2×D20).297C22, C4⋊Dic5.202C22, (Q8×C10).223C22, (C2×Dic5).55C23, C22.144(C23×D5), (C2×Dic10).33C22, C10.D4.69C22, (C22×D5).190C23, D10⋊C4.103C22, (D5×C4⋊C4)⋊18C2, C2.14(C2×Q8×D5), C2.62(C2×C4○D20), C10.55(C2×C4○D4), (C2×C4×D5).83C22, (C5×C4⋊C4).351C22, (C2×C4).169(C22×D5), SmallGroup(320,1251)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D2010Q8
C1C5C10C2×C10C22×D5C2×D20C4×D20 — D2010Q8
C5C2×C10 — D2010Q8
C1C22C4×Q8

Generators and relations for D2010Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd-1=c-1 >

Subgroups: 790 in 228 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×2], Dic5 [×6], C20 [×4], C20 [×5], D10 [×4], D10 [×4], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], D43Q8, C10.D4 [×8], C4⋊Dic5, C4⋊Dic5 [×4], D10⋊C4 [×6], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5 [×6], C2×D20, Q8×C10, C202Q8, C20.6Q8 [×2], C4×D20, C4×D20 [×2], D5×C4⋊C4 [×2], D10⋊Q8 [×4], D103Q8 [×2], Q8×C20, D2010Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ 1+4, C22×D5 [×7], D43Q8, C4○D20 [×2], Q8×D5 [×2], C23×D5, C2×C4○D20, C2×Q8×D5, D48D10, D2010Q8

Smallest permutation representation of D2010Q8
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 23)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(121 123)(124 140)(125 139)(126 138)(127 137)(128 136)(129 135)(130 134)(131 133)(141 143)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)
(1 40 94 107)(2 21 95 108)(3 22 96 109)(4 23 97 110)(5 24 98 111)(6 25 99 112)(7 26 100 113)(8 27 81 114)(9 28 82 115)(10 29 83 116)(11 30 84 117)(12 31 85 118)(13 32 86 119)(14 33 87 120)(15 34 88 101)(16 35 89 102)(17 36 90 103)(18 37 91 104)(19 38 92 105)(20 39 93 106)(41 138 143 76)(42 139 144 77)(43 140 145 78)(44 121 146 79)(45 122 147 80)(46 123 148 61)(47 124 149 62)(48 125 150 63)(49 126 151 64)(50 127 152 65)(51 128 153 66)(52 129 154 67)(53 130 155 68)(54 131 156 69)(55 132 157 70)(56 133 158 71)(57 134 159 72)(58 135 160 73)(59 136 141 74)(60 137 142 75)
(1 53 94 155)(2 54 95 156)(3 55 96 157)(4 56 97 158)(5 57 98 159)(6 58 99 160)(7 59 100 141)(8 60 81 142)(9 41 82 143)(10 42 83 144)(11 43 84 145)(12 44 85 146)(13 45 86 147)(14 46 87 148)(15 47 88 149)(16 48 89 150)(17 49 90 151)(18 50 91 152)(19 51 92 153)(20 52 93 154)(21 69 108 131)(22 70 109 132)(23 71 110 133)(24 72 111 134)(25 73 112 135)(26 74 113 136)(27 75 114 137)(28 76 115 138)(29 77 116 139)(30 78 117 140)(31 79 118 121)(32 80 119 122)(33 61 120 123)(34 62 101 124)(35 63 102 125)(36 64 103 126)(37 65 104 127)(38 66 105 128)(39 67 106 129)(40 68 107 130)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L···4Q5A5B10A···10F20A···20H20I···20AF
order122222224···44444···45510···1020···2020···20
size1111101010102···244420···20222···22···24···4

65 irreducible representations

dim111111112222222444
type++++++++-+++++-+
imageC1C2C2C2C2C2C2C2Q8D5C4○D4D10D10D10C4○D202+ 1+4Q8×D5D48D10
kernelD2010Q8C202Q8C20.6Q8C4×D20D5×C4⋊C4D10⋊Q8D103Q8Q8×C20D20C4×Q8C20C42C4⋊C4C2×Q8C4C10C4C2
# reps1123242142466216144

Matrix representation of D2010Q8 in GL6(𝔽41)

010000
4000000
0014000
0083400
0000400
0000040
,
4000000
010000
0040000
0033100
000010
000001
,
0320000
900000
001000
000100
0000040
000010
,
4000000
0400000
001000
000100
0000320
00002038

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,8,0,0,0,0,40,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,20,0,0,0,0,20,38] >;

D2010Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{10}Q_8
% in TeX

G:=Group("D20:10Q8");
// GroupNames label

G:=SmallGroup(320,1251);
// by ID

G=gap.SmallGroup(320,1251);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,185,192,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽