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G = Dic10⋊D4order 320 = 26·5

6th semidirect product of Dic10 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic106D4, (C5×D4)⋊6D4, (C2×D8)⋊6D5, C4.60(D4×D5), (C10×D8)⋊14C2, C202D45C2, C55(D4⋊D4), D43(C5⋊D4), (C2×C8).35D10, (C2×D4).65D10, C20.167(C2×D4), D101C828C2, C10.56C22≀C2, C10.35(C4○D8), D4⋊Dic530C2, (C22×D5).43D4, C22.258(D4×D5), C2.30(D8⋊D5), C2.19(D83D5), C20.44D429C2, C10.51(C8⋊C22), (C2×C20).435C23, (C2×C40).249C22, (C2×Dic5).237D4, (D4×C10).84C22, C2.24(C23⋊D10), C4⋊Dic5.166C22, (C2×Dic10).126C22, C4.37(C2×C5⋊D4), (C2×D42D5)⋊2C2, (C2×D4.D5)⋊19C2, (C2×C4×D5).49C22, (C2×C10).348(C2×D4), (C2×C4).525(C22×D5), (C2×C52C8).149C22, SmallGroup(320,785)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10⋊D4
C1C5C10C2×C10C2×C20C2×C4×D5C2×D42D5 — Dic10⋊D4
C5C10C2×C20 — Dic10⋊D4
C1C22C2×C4C2×D8

Generators and relations for Dic10⋊D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, dad=a9, cbc-1=a5b, dbd=a10b, dcd=c-1 >

Subgroups: 686 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4, C2×C4 [×9], D4 [×2], D4 [×9], Q8 [×3], C23 [×3], D5, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8 [×2], SD16 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×4], C20 [×2], D10 [×3], C2×C10, C2×C10 [×7], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8, C40, Dic10 [×2], Dic10, C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20, C5×D4 [×2], C5×D4 [×3], C22×D5, C22×C10 [×2], D4⋊D4, C2×C52C8, C4⋊Dic5, D4.D5 [×2], C23.D5, C2×C40, C5×D8 [×2], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4×C10 [×2], C20.44D4, D101C8, D4⋊Dic5, C2×D4.D5, C202D4, C10×D8, C2×D42D5, Dic10⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4⋊D4, D4×D5 [×2], C2×C5⋊D4, D8⋊D5, D83D5, C23⋊D10, Dic10⋊D4

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G···10N20A20B20C20D40A···40H
order12222222444444455888810···1010···102020202040···40
size111144820221010202040224420202···28···844444···4

47 irreducible representations

dim1111111122222222244444
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10C4○D8C5⋊D4C8⋊C22D4×D5D4×D5D8⋊D5D83D5
kernelDic10⋊D4C20.44D4D101C8D4⋊Dic5C2×D4.D5C202D4C10×D8C2×D42D5Dic10C2×Dic5C5×D4C22×D5C2×D8C2×C8C2×D4C10D4C10C4C22C2C2
# reps1111111121212244812244

Matrix representation of Dic10⋊D4 in GL6(𝔽41)

710000
33400000
0040000
0004000
0000940
0000032
,
760000
33340000
0040000
0026100
00003826
0000283
,
34350000
870000
0035900
005600
0000320
000029
,
34350000
870000
0040000
0026100
000019
0000040

G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,40,32],[7,33,0,0,0,0,6,34,0,0,0,0,0,0,40,26,0,0,0,0,0,1,0,0,0,0,0,0,38,28,0,0,0,0,26,3],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,35,5,0,0,0,0,9,6,0,0,0,0,0,0,32,2,0,0,0,0,0,9],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,40,26,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40] >;

Dic10⋊D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,785);
// by ID

G=gap.SmallGroup(320,785);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

׿
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