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G = Dic109D4order 320 = 26·5

2nd semidirect product of Dic10 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C203SD16, Dic109D4, C42.75D10, C4.54(D4×D5), C42(D4.D5), C203C832C2, C54(C4⋊SD16), C41D4.5D5, C20.33(C2×D4), (C2×D4).58D10, (C2×C20).149D4, (C4×Dic10)⋊22C2, C20.77(C4○D4), D4⋊Dic523C2, C4.4(D42D5), C10.58(C2×SD16), C2.13(C202D4), C10.96(C8⋊C22), (C4×C20).123C22, (C2×C20).393C23, (D4×C10).74C22, C10.104(C4⋊D4), C4⋊Dic5.345C22, C2.17(D4.D10), (C2×Dic10).282C22, (C2×D4.D5)⋊14C2, (C5×C41D4).4C2, C2.12(C2×D4.D5), (C2×C10).524(C2×D4), (C2×C4).186(C5⋊D4), (C2×C4).491(C22×D5), C22.197(C2×C5⋊D4), (C2×C52C8).132C22, SmallGroup(320,702)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic109D4
C1C5C10C20C2×C20C2×Dic10C4×Dic10 — Dic109D4
C5C10C2×C20 — Dic109D4
C1C22C42C41D4

Generators and relations for Dic109D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a11, bc=cb, dbd=a5b, dcd=c-1 >

Subgroups: 438 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], C10 [×3], C10 [×2], C42, C42, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], C20, C2×C10, C2×C10 [×6], D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C5×D4 [×8], C22×C10 [×2], C4⋊SD16, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, D4.D5 [×4], C4×C20, C2×Dic10, D4×C10 [×2], D4×C10 [×2], C203C8, D4⋊Dic5 [×2], C4×Dic10, C2×D4.D5 [×2], C5×C41D4, Dic109D4
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, C5⋊D4 [×2], C22×D5, C4⋊SD16, D4.D5 [×2], D4×D5, D42D5, C2×C5⋊D4, D4.D10, C2×D4.D5, C202D4, Dic109D4

Smallest permutation representation of Dic109D4
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 68 11 78)(2 67 12 77)(3 66 13 76)(4 65 14 75)(5 64 15 74)(6 63 16 73)(7 62 17 72)(8 61 18 71)(9 80 19 70)(10 79 20 69)(21 149 31 159)(22 148 32 158)(23 147 33 157)(24 146 34 156)(25 145 35 155)(26 144 36 154)(27 143 37 153)(28 142 38 152)(29 141 39 151)(30 160 40 150)(41 113 51 103)(42 112 52 102)(43 111 53 101)(44 110 54 120)(45 109 55 119)(46 108 56 118)(47 107 57 117)(48 106 58 116)(49 105 59 115)(50 104 60 114)(81 126 91 136)(82 125 92 135)(83 124 93 134)(84 123 94 133)(85 122 95 132)(86 121 96 131)(87 140 97 130)(88 139 98 129)(89 138 99 128)(90 137 100 127)
(1 81 48 24)(2 82 49 25)(3 83 50 26)(4 84 51 27)(5 85 52 28)(6 86 53 29)(7 87 54 30)(8 88 55 31)(9 89 56 32)(10 90 57 33)(11 91 58 34)(12 92 59 35)(13 93 60 36)(14 94 41 37)(15 95 42 38)(16 96 43 39)(17 97 44 40)(18 98 45 21)(19 99 46 22)(20 100 47 23)(61 139 119 159)(62 140 120 160)(63 121 101 141)(64 122 102 142)(65 123 103 143)(66 124 104 144)(67 125 105 145)(68 126 106 146)(69 127 107 147)(70 128 108 148)(71 129 109 149)(72 130 110 150)(73 131 111 151)(74 132 112 152)(75 133 113 153)(76 134 114 154)(77 135 115 155)(78 136 116 156)(79 137 117 157)(80 138 118 158)
(2 12)(4 14)(6 16)(8 18)(10 20)(21 88)(22 99)(23 90)(24 81)(25 92)(26 83)(27 94)(28 85)(29 96)(30 87)(31 98)(32 89)(33 100)(34 91)(35 82)(36 93)(37 84)(38 95)(39 86)(40 97)(41 51)(43 53)(45 55)(47 57)(49 59)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)(101 106)(102 117)(103 108)(104 119)(105 110)(107 112)(109 114)(111 116)(113 118)(115 120)(121 146)(122 157)(123 148)(124 159)(125 150)(126 141)(127 152)(128 143)(129 154)(130 145)(131 156)(132 147)(133 158)(134 149)(135 160)(136 151)(137 142)(138 153)(139 144)(140 155)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G···10N20A···20L
order12222244444444455888810···1010···1020···20
size111188222242020202022202020202···28···84···4

47 irreducible representations

dim1111112222222244444
type++++++++++++-+-
imageC1C2C2C2C2C2D4D4D5SD16C4○D4D10D10C5⋊D4C8⋊C22D4.D5D4×D5D42D5D4.D10
kernelDic109D4C203C8D4⋊Dic5C4×Dic10C2×D4.D5C5×C41D4Dic10C2×C20C41D4C20C20C42C2×D4C2×C4C10C4C4C4C2
# reps1121212224224814224

Matrix representation of Dic109D4 in GL6(𝔽41)

1370000
21400000
0031000
009400
000010
000001
,
0220000
1300000
00192500
0022200
0000400
0000040
,
4000000
0400000
0040000
0004000
00001618
00002925
,
100000
21400000
001000
00284000
000010
00002140

G:=sub<GL(6,GF(41))| [1,21,0,0,0,0,37,40,0,0,0,0,0,0,31,9,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,22,0,0,0,0,0,0,0,19,2,0,0,0,0,25,22,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,29,0,0,0,0,18,25],[1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,21,0,0,0,0,0,40] >;

Dic109D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_9D_4
% in TeX

G:=Group("Dic10:9D4");
// GroupNames label

G:=SmallGroup(320,702);
// by ID

G=gap.SmallGroup(320,702);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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