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G = Dic108D4order 320 = 26·5

1st semidirect product of Dic10 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C202SD16, Dic108D4, C42.39D10, C4⋊C810D5, C42(C40⋊C2), C52(C4⋊SD16), C4.133(D4×D5), C204D4.6C2, (C2×C8).133D10, (C2×C20).124D4, (C2×C4).135D20, C20.342(C2×D4), D205C413C2, (C4×Dic10)⋊18C2, (C4×C20).74C22, C10.12(C2×SD16), C20.331(C4○D4), C10.41(C4⋊D4), C2.20(C8⋊D10), C2.14(C4⋊D20), C10.17(C8⋊C22), (C2×C40).140C22, (C2×C20).758C23, C4.47(Q82D5), (C2×D20).19C22, C22.121(C2×D20), C4⋊Dic5.276C22, (C2×Dic10).221C22, (C5×C4⋊C8)⋊12C2, (C2×C40⋊C2)⋊20C2, C2.15(C2×C40⋊C2), (C2×C10).141(C2×D4), (C2×C4).703(C22×D5), SmallGroup(320,475)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic108D4
C1C5C10C20C2×C20C2×Dic10C4×Dic10 — Dic108D4
C5C10C2×C20 — Dic108D4
C1C22C42C4⋊C8

Generators and relations for Dic108D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a5b, dcd=c-1 >

Subgroups: 710 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×3], C23 [×2], D5 [×2], C10 [×3], C42, C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C2×D4 [×4], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], C20, D10 [×6], C2×C10, D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C40 [×2], Dic10 [×2], Dic10, D20 [×8], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4⋊SD16, C40⋊C2 [×4], C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40 [×2], C2×Dic10, C2×D20 [×2], C2×D20 [×2], D205C4 [×2], C5×C4⋊C8, C4×Dic10, C204D4, C2×C40⋊C2 [×2], Dic108D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C8⋊C22, D20 [×2], C22×D5, C4⋊SD16, C40⋊C2 [×2], C2×D20, D4×D5, Q82D5, C4⋊D20, C2×C40⋊C2, C8⋊D10, Dic108D4

Smallest permutation representation of Dic108D4
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 43 31 53)(22 42 32 52)(23 41 33 51)(24 60 34 50)(25 59 35 49)(26 58 36 48)(27 57 37 47)(28 56 38 46)(29 55 39 45)(30 54 40 44)(61 134 71 124)(62 133 72 123)(63 132 73 122)(64 131 74 121)(65 130 75 140)(66 129 76 139)(67 128 77 138)(68 127 78 137)(69 126 79 136)(70 125 80 135)(101 157 111 147)(102 156 112 146)(103 155 113 145)(104 154 114 144)(105 153 115 143)(106 152 116 142)(107 151 117 141)(108 150 118 160)(109 149 119 159)(110 148 120 158)
(1 139 44 153)(2 140 45 154)(3 121 46 155)(4 122 47 156)(5 123 48 157)(6 124 49 158)(7 125 50 159)(8 126 51 160)(9 127 52 141)(10 128 53 142)(11 129 54 143)(12 130 55 144)(13 131 56 145)(14 132 57 146)(15 133 58 147)(16 134 59 148)(17 135 60 149)(18 136 41 150)(19 137 42 151)(20 138 43 152)(21 106 94 77)(22 107 95 78)(23 108 96 79)(24 109 97 80)(25 110 98 61)(26 111 99 62)(27 112 100 63)(28 113 81 64)(29 114 82 65)(30 115 83 66)(31 116 84 67)(32 117 85 68)(33 118 86 69)(34 119 87 70)(35 120 88 71)(36 101 89 72)(37 102 90 73)(38 103 91 74)(39 104 92 75)(40 105 93 76)
(1 139)(2 138)(3 137)(4 136)(5 135)(6 134)(7 133)(8 132)(9 131)(10 130)(11 129)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 121)(20 140)(21 119)(22 118)(23 117)(24 116)(25 115)(26 114)(27 113)(28 112)(29 111)(30 110)(31 109)(32 108)(33 107)(34 106)(35 105)(36 104)(37 103)(38 102)(39 101)(40 120)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 160)(58 159)(59 158)(60 157)(61 83)(62 82)(63 81)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12222244444444455888810···1020···2020···2040···40
size1111404022224202020202244442···22···24···44···4

59 irreducible representations

dim1111112222222224444
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D5SD16C4○D4D10D10D20C40⋊C2C8⋊C22D4×D5Q82D5C8⋊D10
kernelDic108D4D205C4C5×C4⋊C8C4×Dic10C204D4C2×C40⋊C2Dic10C2×C20C4⋊C8C20C20C42C2×C8C2×C4C4C10C4C4C2
# reps12111222242248161224

Matrix representation of Dic108D4 in GL6(𝔽41)

190000
18400000
0004000
0013500
0000400
0000040
,
0120000
1700000
0021300
00253900
000014
0000040
,
4000000
0400000
0040000
0004000
0000325
000009
,
4000000
2310000
0040000
0035100
0000936
00001632

G:=sub<GL(6,GF(41))| [1,18,0,0,0,0,9,40,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,2,25,0,0,0,0,13,39,0,0,0,0,0,0,1,0,0,0,0,0,4,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,5,9],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,9,16,0,0,0,0,36,32] >;

Dic108D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_8D_4
% in TeX

G:=Group("Dic10:8D4");
// GroupNames label

G:=SmallGroup(320,475);
// by ID

G=gap.SmallGroup(320,475);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽