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G = Dic1020D4order 320 = 26·5

8th semidirect product of Dic10 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1020D4, C10.332+ 1+4, C4⋊D47D5, C53(Q86D4), C4.109(D4×D5), C20⋊D415C2, C4⋊D2019C2, C4⋊C4.177D10, (C2×D4).90D10, C20.225(C2×D4), D10⋊D417C2, Dic58(C4○D4), Dic54D46C2, (C2×C20).35C23, C22⋊C4.46D10, Dic5.45(C2×D4), C10.62(C22×D4), Dic53Q820C2, Dic5⋊D410C2, (C2×C10).143C24, (C22×C4).219D10, C2.35(D46D10), C23.10(C22×D5), (C2×D20).148C22, (D4×C10).117C22, (C22×C10).14C23, (C4×Dic5).98C22, (C22×D5).62C23, C22.164(C23×D5), D10⋊C4.12C22, (C22×C20).237C22, (C2×Dic5).235C23, C10.D4.14C22, C23.D5.110C22, (C2×Dic10).301C22, (C22×Dic5).104C22, C2.35(C2×D4×D5), (C5×C4⋊D4)⋊8C2, (C4×C5⋊D4)⋊15C2, C2.34(D5×C4○D4), (C2×C4○D20)⋊19C2, (C2×D42D5)⋊11C2, (C2×C4×D5).91C22, C10.148(C2×C4○D4), (C5×C4⋊C4).139C22, (C2×C4).585(C22×D5), (C2×C5⋊D4).25C22, (C5×C22⋊C4).8C22, SmallGroup(320,1271)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic1020D4
C1C5C10C2×C10C2×Dic5C2×Dic10C2×C4○D20 — Dic1020D4
C5C2×C10 — Dic1020D4
C1C22C4⋊D4

Generators and relations for Dic1020D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1222 in 312 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×18], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×24], Q8 [×4], C23, C23 [×2], C23 [×3], D5 [×3], C10 [×3], C10 [×3], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×12], C2×Q8, C4○D4 [×8], Dic5 [×6], Dic5 [×2], C20 [×2], C20 [×3], D10 [×9], C2×C10, C2×C10 [×9], C4×D4 [×3], C4×Q8, C4⋊D4, C4⋊D4 [×5], C41D4 [×3], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×6], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×2], C22×C10, C22×C10 [×2], Q86D4, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], D10⋊C4, D10⋊C4 [×2], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C4○D20 [×4], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4, C2×C5⋊D4 [×8], C22×C20, D4×C10, D4×C10 [×2], Dic54D4 [×2], D10⋊D4 [×2], Dic53Q8, C4⋊D20, C4×C5⋊D4, Dic5⋊D4 [×2], C20⋊D4, C20⋊D4 [×2], C5×C4⋊D4, C2×C4○D20, C2×D42D5, Dic1020D4
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], Q86D4, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10, D5×C4○D4, Dic1020D4

Smallest permutation representation of Dic1020D4
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)
(1 115 11 105)(2 114 12 104)(3 113 13 103)(4 112 14 102)(5 111 15 101)(6 110 16 120)(7 109 17 119)(8 108 18 118)(9 107 19 117)(10 106 20 116)(21 132 31 122)(22 131 32 121)(23 130 33 140)(24 129 34 139)(25 128 35 138)(26 127 36 137)(27 126 37 136)(28 125 38 135)(29 124 39 134)(30 123 40 133)(41 158 51 148)(42 157 52 147)(43 156 53 146)(44 155 54 145)(45 154 55 144)(46 153 56 143)(47 152 57 142)(48 151 58 141)(49 150 59 160)(50 149 60 159)(61 100 71 90)(62 99 72 89)(63 98 73 88)(64 97 74 87)(65 96 75 86)(66 95 76 85)(67 94 77 84)(68 93 78 83)(69 92 79 82)(70 91 80 81)
(1 91 39 52)(2 82 40 43)(3 93 21 54)(4 84 22 45)(5 95 23 56)(6 86 24 47)(7 97 25 58)(8 88 26 49)(9 99 27 60)(10 90 28 51)(11 81 29 42)(12 92 30 53)(13 83 31 44)(14 94 32 55)(15 85 33 46)(16 96 34 57)(17 87 35 48)(18 98 36 59)(19 89 37 50)(20 100 38 41)(61 135 148 116)(62 126 149 107)(63 137 150 118)(64 128 151 109)(65 139 152 120)(66 130 153 111)(67 121 154 102)(68 132 155 113)(69 123 156 104)(70 134 157 115)(71 125 158 106)(72 136 159 117)(73 127 160 108)(74 138 141 119)(75 129 142 110)(76 140 143 101)(77 131 144 112)(78 122 145 103)(79 133 146 114)(80 124 147 105)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 61)(59 62)(60 63)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 141)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 150)(100 151)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4N4O5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222222224444444···445510···10101010101010101020···2020202020
size111144420202022224410···1020222···2444488884···48888

53 irreducible representations

dim1111111111122222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D102+ 1+4D4×D5D46D10D5×C4○D4
kernelDic1020D4Dic54D4D10⋊D4Dic53Q8C4⋊D20C4×C5⋊D4Dic5⋊D4C20⋊D4C5×C4⋊D4C2×C4○D20C2×D42D5Dic10C4⋊D4Dic5C22⋊C4C4⋊C4C22×C4C2×D4C10C4C2C2
# reps1221112311142442261444

Matrix representation of Dic1020D4 in GL6(𝔽41)

4000000
0400000
0022200
00241900
0000740
000010
,
4000000
0400000
009000
0073200
0000734
0000134
,
1400000
2400000
0072300
00213400
000010
000001
,
100000
2400000
0072300
00303400
000010
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,24,0,0,0,0,2,19,0,0,0,0,0,0,7,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,7,0,0,0,0,0,32,0,0,0,0,0,0,7,1,0,0,0,0,34,34],[1,2,0,0,0,0,40,40,0,0,0,0,0,0,7,21,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,7,30,0,0,0,0,23,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic1020D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{20}D_4
% in TeX

G:=Group("Dic10:20D4");
// GroupNames label

G:=SmallGroup(320,1271);
// by ID

G=gap.SmallGroup(320,1271);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,184,570,185,12550]);
// 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,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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