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G = D208Q8order 320 = 26·5

6th semidirect product of D20 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D208Q8, Dic1012D4, C42.173D10, C10.362- 1+4, C54(D4×Q8), C41(Q8×D5), C4⋊Q811D5, C203(C2×Q8), C20⋊Q844C2, D107(C2×Q8), C4.74(D4×D5), C20.72(C2×D4), C4⋊C4.218D10, (C4×D20).26C2, D10⋊Q848C2, D103Q836C2, (C4×Dic10)⋊52C2, (C2×Q8).146D10, Dic5.54(C2×D4), D208C4.13C2, C10.47(C22×Q8), (C2×C20).104C23, (C4×C20).212C22, (C2×C10).271C24, C10.101(C22×D4), (C2×D20).280C22, C4⋊Dic5.385C22, (Q8×C10).138C22, C22.292(C23×D5), (C4×Dic5).168C22, (C2×Dic5).142C23, (C22×D5).242C23, D10⋊C4.152C22, C2.37(Q8.10D10), (C2×Dic10).195C22, C10.D4.166C22, (C2×Q8×D5)⋊13C2, C2.74(C2×D4×D5), C2.30(C2×Q8×D5), (C5×C4⋊Q8)⋊13C2, (C2×C4×D5).154C22, (C5×C4⋊C4).214C22, (C2×C4).218(C22×D5), SmallGroup(320,1399)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D208Q8
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q8×D5 — D208Q8
C5C2×C10 — D208Q8
C1C22C4⋊Q8

Generators and relations for D208Q8
 G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 966 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C4⋊Q8, C22×Q8, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, D4×Q8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, Q8×D5, Q8×C10, C4×Dic10, C4×D20, C20⋊Q8, D208C4, D10⋊Q8, D103Q8, C5×C4⋊Q8, C2×Q8×D5, D208Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C24, D10, C22×D4, C22×Q8, 2- 1+4, C22×D5, D4×Q8, D4×D5, Q8×D5, C23×D5, C2×D4×D5, C2×Q8×D5, Q8.10D10, D208Q8

Smallest permutation representation of D208Q8
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 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 109)(102 108)(103 107)(104 106)(110 120)(111 119)(112 118)(113 117)(114 116)(121 127)(122 126)(123 125)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)
(1 76 40 50)(2 77 21 51)(3 78 22 52)(4 79 23 53)(5 80 24 54)(6 61 25 55)(7 62 26 56)(8 63 27 57)(9 64 28 58)(10 65 29 59)(11 66 30 60)(12 67 31 41)(13 68 32 42)(14 69 33 43)(15 70 34 44)(16 71 35 45)(17 72 36 46)(18 73 37 47)(19 74 38 48)(20 75 39 49)(81 131 145 102)(82 132 146 103)(83 133 147 104)(84 134 148 105)(85 135 149 106)(86 136 150 107)(87 137 151 108)(88 138 152 109)(89 139 153 110)(90 140 154 111)(91 121 155 112)(92 122 156 113)(93 123 157 114)(94 124 158 115)(95 125 159 116)(96 126 160 117)(97 127 141 118)(98 128 142 119)(99 129 143 120)(100 130 144 101)
(1 113 40 122)(2 104 21 133)(3 115 22 124)(4 106 23 135)(5 117 24 126)(6 108 25 137)(7 119 26 128)(8 110 27 139)(9 101 28 130)(10 112 29 121)(11 103 30 132)(12 114 31 123)(13 105 32 134)(14 116 33 125)(15 107 34 136)(16 118 35 127)(17 109 36 138)(18 120 37 129)(19 111 38 140)(20 102 39 131)(41 93 67 157)(42 84 68 148)(43 95 69 159)(44 86 70 150)(45 97 71 141)(46 88 72 152)(47 99 73 143)(48 90 74 154)(49 81 75 145)(50 92 76 156)(51 83 77 147)(52 94 78 158)(53 85 79 149)(54 96 80 160)(55 87 61 151)(56 98 62 142)(57 89 63 153)(58 100 64 144)(59 91 65 155)(60 82 66 146)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4I4J4K4L4M4N4O4P4Q5A5B10A···10F20A···20L20M···20T
order1222222244444···4444444445510···1020···2020···20
size11111010101022224···41010101020202020222···24···48···8

53 irreducible representations

dim1111111112222224444
type++++++++++-++++-+-
imageC1C2C2C2C2C2C2C2C2D4Q8D5D10D10D102- 1+4D4×D5Q8×D5Q8.10D10
kernelD208Q8C4×Dic10C4×D20C20⋊Q8D208C4D10⋊Q8D103Q8C5×C4⋊Q8C2×Q8×D5Dic10D20C4⋊Q8C42C4⋊C4C2×Q8C10C4C4C2
# reps1112242124422841444

Matrix representation of D208Q8 in GL6(𝔽41)

840000
35330000
0035100
0054000
0000400
0000040
,
100000
37400000
00404000
000100
000010
000001
,
100000
010000
0040000
0004000
0000040
000010
,
33370000
2680000
0040000
0004000
0000301
0000111

G:=sub<GL(6,GF(41))| [8,35,0,0,0,0,4,33,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,37,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[33,26,0,0,0,0,37,8,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,1,0,0,0,0,1,11] >;

D208Q8 in GAP, Magma, Sage, TeX

D_{20}\rtimes_8Q_8
% in TeX

G:=Group("D20:8Q8");
// GroupNames label

G:=SmallGroup(320,1399);
// by ID

G=gap.SmallGroup(320,1399);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,136,12550]);
// 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,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

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