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G = Dic1023D4order 320 = 26·5

1st semidirect product of Dic10 and D4 acting through Inn(Dic10)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1023D4, C42.110D10, C10.602- 1+4, (C4×D4)⋊14D5, (D4×C20)⋊16C2, C207D48C2, C51(Q85D4), C4.141(D4×D5), C4⋊C4.283D10, D10⋊Q87C2, C20.347(C2×D4), (C4×Dic10)⋊31C2, (C2×D4).215D10, C4.D2017C2, C221(C4○D20), (C2×C10).96C24, Dic5.41(C2×D4), C10.51(C22×D4), Dic54D447C2, Dic5⋊D426C2, C20.48D421C2, (C4×C20).153C22, (C2×C20).784C23, C22⋊C4.111D10, Dic5.5D46C2, (C22×Dic10)⋊9C2, (C22×C4).209D10, C23.96(C22×D5), (C2×D20).219C22, (D4×C10).306C22, C4⋊Dic5.298C22, (C22×D5).31C23, C22.121(C23×D5), C23.D5.13C22, D10⋊C4.66C22, (C22×C20).108C22, (C22×C10).166C23, (C2×Dic5).214C23, (C4×Dic5).224C22, C2.17(D4.10D10), (C2×Dic10).248C22, C10.D4.154C22, (C22×Dic5).96C22, C2.24(C2×D4×D5), (C2×C4○D20)⋊9C2, (C2×C10)⋊3(C4○D4), C10.43(C2×C4○D4), C2.47(C2×C4○D20), (C2×C4×D5).250C22, (C5×C4⋊C4).327C22, (C2×C4).159(C22×D5), (C2×C5⋊D4).122C22, (C5×C22⋊C4).123C22, SmallGroup(320,1224)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic1023D4
C1C5C10C2×C10C22×D5C2×C5⋊D4Dic54D4 — Dic1023D4
C5C2×C10 — Dic1023D4
C1C22C4×D4

Generators and relations for Dic1023D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd=c-1 >

Subgroups: 1030 in 290 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×5], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×5], C2×Q8 [×8], C4○D4 [×4], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×5], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×6], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], Q85D4, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×2], C2×Dic10 [×4], C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C4×Dic10, C4.D20, Dic54D4 [×2], Dic5.5D4 [×2], D10⋊Q8 [×2], C20.48D4, C207D4, Dic5⋊D4 [×2], D4×C20, C22×Dic10, C2×C4○D20, Dic1023D4
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, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4.10D10, Dic1023D4

Smallest permutation representation of Dic1023D4
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 54 11 44)(2 53 12 43)(3 52 13 42)(4 51 14 41)(5 50 15 60)(6 49 16 59)(7 48 17 58)(8 47 18 57)(9 46 19 56)(10 45 20 55)(21 124 31 134)(22 123 32 133)(23 122 33 132)(24 121 34 131)(25 140 35 130)(26 139 36 129)(27 138 37 128)(28 137 38 127)(29 136 39 126)(30 135 40 125)(61 81 71 91)(62 100 72 90)(63 99 73 89)(64 98 74 88)(65 97 75 87)(66 96 76 86)(67 95 77 85)(68 94 78 84)(69 93 79 83)(70 92 80 82)(101 149 111 159)(102 148 112 158)(103 147 113 157)(104 146 114 156)(105 145 115 155)(106 144 116 154)(107 143 117 153)(108 142 118 152)(109 141 119 151)(110 160 120 150)
(1 156 80 28)(2 157 61 29)(3 158 62 30)(4 159 63 31)(5 160 64 32)(6 141 65 33)(7 142 66 34)(8 143 67 35)(9 144 68 36)(10 145 69 37)(11 146 70 38)(12 147 71 39)(13 148 72 40)(14 149 73 21)(15 150 74 22)(16 151 75 23)(17 152 76 24)(18 153 77 25)(19 154 78 26)(20 155 79 27)(41 101 89 134)(42 102 90 135)(43 103 91 136)(44 104 92 137)(45 105 93 138)(46 106 94 139)(47 107 95 140)(48 108 96 121)(49 109 97 122)(50 110 98 123)(51 111 99 124)(52 112 100 125)(53 113 81 126)(54 114 82 127)(55 115 83 128)(56 116 84 129)(57 117 85 130)(58 118 86 131)(59 119 87 132)(60 120 88 133)
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 101)(68 102)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10F10G···10N20A···20H20I···20X
order1222222224···444444444445510···1010···1020···2020···20
size111122420202···2441010101020202020222···24···42···24···4

65 irreducible representations

dim111111111111222222222444
type+++++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D10D10C4○D202- 1+4D4×D5D4.10D10
kernelDic1023D4C4×Dic10C4.D20Dic54D4Dic5.5D4D10⋊Q8C20.48D4C207D4Dic5⋊D4D4×C20C22×Dic10C2×C4○D20Dic10C4×D4C2×C10C42C22⋊C4C4⋊C4C22×C4C2×D4C22C10C4C2
# reps1112221121114242424216144

Matrix representation of Dic1023D4 in GL4(𝔽41) generated by

111600
392700
0010
0001
,
382000
20300
00400
00040
,
243400
61700
0012
004040
,
1000
0100
004039
0001
G:=sub<GL(4,GF(41))| [11,39,0,0,16,27,0,0,0,0,1,0,0,0,0,1],[38,20,0,0,20,3,0,0,0,0,40,0,0,0,0,40],[24,6,0,0,34,17,0,0,0,0,1,40,0,0,2,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,39,1] >;

Dic1023D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{23}D_4
% in TeX

G:=Group("Dic10:23D4");
// GroupNames label

G:=SmallGroup(320,1224);
// by ID

G=gap.SmallGroup(320,1224);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,100,675,570,12550]);
// 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,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽