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G = Dic10.11D4order 320 = 26·5

11st non-split extension by Dic10 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10.11D4, C4.91(D4×D5), Q8⋊C49D5, C4⋊C4.146D10, C4.6(C4○D20), C20.115(C2×D4), (C2×C8).120D10, (C2×Q8).13D10, C52(Q8.D4), Dic53Q85C2, D206C4.2C2, C10.45(C4○D8), C20.17(C4○D4), C20.8Q813C2, (C2×Dic5).36D4, C22.192(D4×D5), C10.22(C4⋊D4), (C2×C20).238C23, (C2×C40).131C22, C20.23D4.5C2, (C2×D20).62C22, (Q8×C10).21C22, C2.25(D10⋊D4), C2.11(Q16⋊D5), C10.56(C8.C22), (C4×Dic5).28C22, C2.14(SD163D5), (C2×Dic10).71C22, (C2×C5⋊Q16)⋊2C2, (C5×Q8⋊C4)⋊9C2, (C2×C40⋊C2).3C2, (C2×C10).251(C2×D4), (C5×C4⋊C4).39C22, (C2×C52C8).33C22, (C2×C4).345(C22×D5), SmallGroup(320,425)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10.11D4
C1C5C10C20C2×C20C4×Dic5Dic53Q8 — Dic10.11D4
C5C10C2×C20 — Dic10.11D4
C1C22C2×C4Q8⋊C4

Generators and relations for Dic10.11D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, bc=cb, dbd=a15b, dcd=a10c-1 >

Subgroups: 486 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×5], C23, D5, C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], D10 [×3], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8, C40, Dic10 [×2], Dic10, D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, Q8.D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4 [×2], C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, Q8×C10, D206C4, C20.8Q8, C5×Q8⋊C4, Dic53Q8, C2×C40⋊C2, C2×C5⋊Q16, C20.23D4, Dic10.11D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8.C22, C22×D5, Q8.D4, C4○D20, D4×D5 [×2], D10⋊D4, SD163D5, Q16⋊D5, Dic10.11D4

Smallest permutation representation of Dic10.11D4
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 30 11 40)(2 29 12 39)(3 28 13 38)(4 27 14 37)(5 26 15 36)(6 25 16 35)(7 24 17 34)(8 23 18 33)(9 22 19 32)(10 21 20 31)(41 110 51 120)(42 109 52 119)(43 108 53 118)(44 107 54 117)(45 106 55 116)(46 105 56 115)(47 104 57 114)(48 103 58 113)(49 102 59 112)(50 101 60 111)(61 93 71 83)(62 92 72 82)(63 91 73 81)(64 90 74 100)(65 89 75 99)(66 88 76 98)(67 87 77 97)(68 86 78 96)(69 85 79 95)(70 84 80 94)(121 148 131 158)(122 147 132 157)(123 146 133 156)(124 145 134 155)(125 144 135 154)(126 143 136 153)(127 142 137 152)(128 141 138 151)(129 160 139 150)(130 159 140 149)
(1 100 55 143)(2 89 56 152)(3 98 57 141)(4 87 58 150)(5 96 59 159)(6 85 60 148)(7 94 41 157)(8 83 42 146)(9 92 43 155)(10 81 44 144)(11 90 45 153)(12 99 46 142)(13 88 47 151)(14 97 48 160)(15 86 49 149)(16 95 50 158)(17 84 51 147)(18 93 52 156)(19 82 53 145)(20 91 54 154)(21 63 107 135)(22 72 108 124)(23 61 109 133)(24 70 110 122)(25 79 111 131)(26 68 112 140)(27 77 113 129)(28 66 114 138)(29 75 115 127)(30 64 116 136)(31 73 117 125)(32 62 118 134)(33 71 119 123)(34 80 120 132)(35 69 101 121)(36 78 102 130)(37 67 103 139)(38 76 104 128)(39 65 105 137)(40 74 106 126)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 134)(62 133)(63 132)(64 131)(65 130)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 140)(76 139)(77 138)(78 137)(79 136)(80 135)(81 152)(82 151)(83 150)(84 149)(85 148)(86 147)(87 146)(88 145)(89 144)(90 143)(91 142)(92 141)(93 160)(94 159)(95 158)(96 157)(97 156)(98 155)(99 154)(100 153)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)(117 120)(118 119)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444444455888810···102020202020···2040···40
size111140224481010202020224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type++++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C4○D20C8.C22D4×D5D4×D5SD163D5Q16⋊D5
kernelDic10.11D4D206C4C20.8Q8C5×Q8⋊C4Dic53Q8C2×C40⋊C2C2×C5⋊Q16C20.23D4Dic10C2×Dic5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C10C4C10C4C22C2C2
# reps1111111122222224812244

Matrix representation of Dic10.11D4 in GL6(𝔽41)

010000
4060000
0040000
0004000
0000402
0000401
,
100000
6400000
0092300
0093200
0000011
0000260
,
100000
6400000
0040200
0040100
000090
000009
,
100000
6400000
001000
0014000
000010
0000140

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,9,9,0,0,0,0,23,32,0,0,0,0,0,0,0,26,0,0,0,0,11,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

Dic10.11D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}._{11}D_4
% in TeX

G:=Group("Dic10.11D4");
// GroupNames label

G:=SmallGroup(320,425);
// by ID

G=gap.SmallGroup(320,425);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,b*c=c*b,d*b*d=a^15*b,d*c*d=a^10*c^-1>;
// generators/relations

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