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G = Dic1010D4order 320 = 26·5

3rd semidirect product of Dic10 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1010D4, C42.142D10, C10.912- 1+4, C4.71(D4×D5), (C4×D20)⋊44C2, C55(Q85D4), C20.64(C2×D4), C202D434C2, C4.4D411D5, D1015(C4○D4), D10⋊D441C2, D103Q829C2, (C4×Dic10)⋊45C2, (C2×D4).174D10, (C2×C20).81C23, (C2×Q8).137D10, C22⋊C4.73D10, Dic5.52(C2×D4), C10.91(C22×D4), Dic54D430C2, (C2×C10).221C24, (C4×C20).186C22, C23.43(C22×D5), Dic5.5D440C2, (D4×C10).156C22, (C2×D20).231C22, C4⋊Dic5.377C22, (C22×C10).51C23, (Q8×C10).127C22, C22.242(C23×D5), Dic5.14D441C2, C23.D5.55C22, (C4×Dic5).234C22, (C2×Dic5).263C23, (C22×D5).226C23, C2.52(D4.10D10), D10⋊C4.135C22, (C2×Dic10).305C22, C10.D4.121C22, (C22×Dic5).143C22, (C2×Q8×D5)⋊11C2, C2.64(C2×D4×D5), C2.77(D5×C4○D4), (C2×D42D5)⋊19C2, C10.188(C2×C4○D4), (C5×C4.4D4)⋊13C2, (C2×C4×D5).130C22, (C2×C4).196(C22×D5), (C2×C5⋊D4).60C22, (C5×C22⋊C4).65C22, SmallGroup(320,1349)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic1010D4
C1C5C10C2×C10C22×D5C2×C4×D5C2×Q8×D5 — Dic1010D4
C5C2×C10 — Dic1010D4
C1C22C4.4D4

Generators and relations for Dic1010D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a9, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1046 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×13], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], D5 [×3], C10 [×3], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×5], C2×C10, C2×C10 [×6], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4, C4.4D4 [×2], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×4], C4×D5 [×8], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], Q85D4, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, D42D5 [×4], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic5.14D4 [×2], Dic54D4 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], C202D4, D103Q8, C5×C4.4D4, C2×D42D5, C2×Q8×D5, Dic1010D4
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], C23×D5, C2×D4×D5, D5×C4○D4, D4.10D10, Dic1010D4

Smallest permutation representation of Dic1010D4
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)
(1 119 11 109)(2 118 12 108)(3 117 13 107)(4 116 14 106)(5 115 15 105)(6 114 16 104)(7 113 17 103)(8 112 18 102)(9 111 19 101)(10 110 20 120)(21 54 31 44)(22 53 32 43)(23 52 33 42)(24 51 34 41)(25 50 35 60)(26 49 36 59)(27 48 37 58)(28 47 38 57)(29 46 39 56)(30 45 40 55)(61 95 71 85)(62 94 72 84)(63 93 73 83)(64 92 74 82)(65 91 75 81)(66 90 76 100)(67 89 77 99)(68 88 78 98)(69 87 79 97)(70 86 80 96)(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 85 121 29)(2 86 122 30)(3 87 123 31)(4 88 124 32)(5 89 125 33)(6 90 126 34)(7 91 127 35)(8 92 128 36)(9 93 129 37)(10 94 130 38)(11 95 131 39)(12 96 132 40)(13 97 133 21)(14 98 134 22)(15 99 135 23)(16 100 136 24)(17 81 137 25)(18 82 138 26)(19 83 139 27)(20 84 140 28)(41 104 76 147)(42 105 77 148)(43 106 78 149)(44 107 79 150)(45 108 80 151)(46 109 61 152)(47 110 62 153)(48 111 63 154)(49 112 64 155)(50 113 65 156)(51 114 66 157)(52 115 67 158)(53 116 68 159)(54 117 69 160)(55 118 70 141)(56 119 71 142)(57 120 72 143)(58 101 73 144)(59 102 74 145)(60 103 75 146)
(1 121)(2 130)(3 139)(4 128)(5 137)(6 126)(7 135)(8 124)(9 133)(10 122)(11 131)(12 140)(13 129)(14 138)(15 127)(16 136)(17 125)(18 134)(19 123)(20 132)(21 37)(22 26)(23 35)(25 33)(27 31)(28 40)(30 38)(32 36)(41 51)(42 60)(43 49)(44 58)(45 47)(46 56)(48 54)(50 52)(53 59)(55 57)(61 71)(62 80)(63 69)(64 78)(65 67)(66 76)(68 74)(70 72)(73 79)(75 77)(81 89)(82 98)(83 87)(84 96)(86 94)(88 92)(91 99)(93 97)(101 150)(102 159)(103 148)(104 157)(105 146)(106 155)(107 144)(108 153)(109 142)(110 151)(111 160)(112 149)(113 158)(114 147)(115 156)(116 145)(117 154)(118 143)(119 152)(120 141)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H···4M4N4O4P5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222222244444444···44445510···101010101020···2020202020
size111144101020222244410···10202020222···288884···48888

53 irreducible representations

dim11111111111122222224444
type++++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D102- 1+4D4×D5D5×C4○D4D4.10D10
kernelDic1010D4C4×Dic10C4×D20Dic5.14D4Dic54D4D10⋊D4Dic5.5D4C202D4D103Q8C5×C4.4D4C2×D42D5C2×Q8×D5Dic10C4.4D4D10C42C22⋊C4C2×D4C2×Q8C10C4C2C2
# reps11122221111142428221444

Matrix representation of Dic1010D4 in GL6(𝔽41)

7400000
100000
001000
000100
000090
00003232
,
3470000
4070000
0040000
0004000
00004039
000011
,
100000
010000
000100
0040000
000010
00004040
,
3470000
4070000
0040000
000100
000010
00004040

G:=sub<GL(6,GF(41))| [7,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,0,32],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;

Dic1010D4 in GAP, Magma, Sage, TeX

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

G:=Group("Dic10:10D4");
// GroupNames label

G:=SmallGroup(320,1349);
// by ID

G=gap.SmallGroup(320,1349);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,192,12550]);
// 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^9,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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