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G = Dic1019D4order 320 = 26·5

7th semidirect product of Dic10 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1019D4, C10.692- 1+4, C4⋊D46D5, C53(Q85D4), C4.108(D4×D5), C4⋊C4.176D10, (D4×Dic5)⋊15C2, C20.224(C2×D4), C22⋊C4.5D10, D102Q819C2, Dic5⋊D49C2, (C2×D4).151D10, Dic5.44(C2×D4), C10.61(C22×D4), Dic53Q819C2, C20.17D414C2, C221(D42D5), C23.9(C22×D5), (C2×C20).500C23, (C2×C10).142C24, (C22×C4).218D10, Dic5.5D417C2, (C22×Dic10)⋊16C2, (D4×C10).116C22, C4⋊Dic5.204C22, (C22×C10).13C23, (C4×Dic5).97C22, (C2×Dic5).65C23, (C22×D5).61C23, C22.163(C23×D5), Dic5.14D416C2, C23.D5.20C22, D10⋊C4.11C22, (C22×C20).236C22, C10.D4.13C22, C2.27(D4.10D10), (C2×Dic10).253C22, (C22×Dic5).103C22, C2.34(C2×D4×D5), (C5×C4⋊D4)⋊7C2, (C4×C5⋊D4)⋊14C2, (C2×C10)⋊4(C4○D4), C10.80(C2×C4○D4), (C2×D42D5)⋊10C2, (C2×C4×D5).90C22, C2.31(C2×D42D5), (C5×C4⋊C4).138C22, (C2×C4).173(C22×D5), (C5×C22⋊C4).7C22, (C2×C5⋊D4).126C22, SmallGroup(320,1270)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic1019D4
C1C5C10C2×C10C2×Dic5C2×Dic10C22×Dic10 — Dic1019D4
C5C2×C10 — Dic1019D4
C1C22C4⋊D4

Generators and relations for Dic1019D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 982 in 290 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×19], D4 [×12], Q8 [×10], C23, C23 [×2], C23, D5, C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×8], C4○D4 [×4], Dic5 [×4], Dic5 [×5], C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×2], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×6], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×8], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×2], Q85D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×C4×D5, D42D5 [×4], C22×Dic5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, D4×C10, D4×C10 [×2], Dic5.14D4 [×2], Dic5.5D4 [×2], Dic53Q8, D102Q8, C4×C5⋊D4, D4×Dic5 [×2], C20.17D4, Dic5⋊D4 [×2], C5×C4⋊D4, C22×Dic10, C2×D42D5, Dic1019D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C22×D5 [×7], Q85D4, D4×D5 [×2], D42D5 [×2], C23×D5, C2×D4×D5, C2×D42D5, D4.10D10, Dic1019D4

Smallest permutation representation of Dic1019D4
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 83 11 93)(2 82 12 92)(3 81 13 91)(4 100 14 90)(5 99 15 89)(6 98 16 88)(7 97 17 87)(8 96 18 86)(9 95 19 85)(10 94 20 84)(21 80 31 70)(22 79 32 69)(23 78 33 68)(24 77 34 67)(25 76 35 66)(26 75 36 65)(27 74 37 64)(28 73 38 63)(29 72 39 62)(30 71 40 61)(41 110 51 120)(42 109 52 119)(43 108 53 118)(44 107 54 117)(45 106 55 116)(46 105 56 115)(47 104 57 114)(48 103 58 113)(49 102 59 112)(50 101 60 111)(121 142 131 152)(122 141 132 151)(123 160 133 150)(124 159 134 149)(125 158 135 148)(126 157 136 147)(127 156 137 146)(128 155 138 145)(129 154 139 144)(130 153 140 143)
(1 36 155 52)(2 27 156 43)(3 38 157 54)(4 29 158 45)(5 40 159 56)(6 31 160 47)(7 22 141 58)(8 33 142 49)(9 24 143 60)(10 35 144 51)(11 26 145 42)(12 37 146 53)(13 28 147 44)(14 39 148 55)(15 30 149 46)(16 21 150 57)(17 32 151 48)(18 23 152 59)(19 34 153 50)(20 25 154 41)(61 134 115 99)(62 125 116 90)(63 136 117 81)(64 127 118 92)(65 138 119 83)(66 129 120 94)(67 140 101 85)(68 131 102 96)(69 122 103 87)(70 133 104 98)(71 124 105 89)(72 135 106 100)(73 126 107 91)(74 137 108 82)(75 128 109 93)(76 139 110 84)(77 130 111 95)(78 121 112 86)(79 132 113 97)(80 123 114 88)
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 136)(30 137)(31 138)(32 139)(33 140)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 97)(42 98)(43 99)(44 100)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 156)(62 157)(63 158)(64 159)(65 160)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F···4M4N4O4P5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order122222222444444···44445510···10101010101010101020···2020202020
size11112244202244410···10202020222···2444488884···48888

53 irreducible representations

dim11111111111122222224444
type++++++++++++++++++-+--
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D102- 1+4D4×D5D42D5D4.10D10
kernelDic1019D4Dic5.14D4Dic5.5D4Dic53Q8D102Q8C4×C5⋊D4D4×Dic5C20.17D4Dic5⋊D4C5×C4⋊D4C22×Dic10C2×D42D5Dic10C4⋊D4C2×C10C22⋊C4C4⋊C4C22×C4C2×D4C10C4C22C2
# reps12211121211142442261444

Matrix representation of Dic1019D4 in GL6(𝔽41)

900000
0320000
000100
00403400
000010
000001
,
090000
900000
001000
00344000
0000400
0000040
,
010000
100000
001000
000100
00002331
00001218
,
100000
010000
001000
000100
00001810
00002123

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,12,0,0,0,0,31,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,21,0,0,0,0,10,23] >;

Dic1019D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{19}D_4
% in TeX

G:=Group("Dic10:19D4");
// GroupNames label

G:=SmallGroup(320,1270);
// by ID

G=gap.SmallGroup(320,1270);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,570,185,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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×
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