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G = C403D4order 320 = 26·5

3rd semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C403D4, C23.18D20, C82(C5⋊D4), C54(C82D4), C406C44C2, (C2×D40)⋊12C2, (C2×C4).52D20, (C2×C8).78D10, C207D441C2, C20.421(C2×D4), (C2×C20).298D4, (C2×M4(2))⋊2D5, D205C442C2, (C10×M4(2))⋊2C2, (C2×C40).64C22, C4.115(C4○D20), C20.231(C4○D4), C2.23(C8⋊D10), C10.74(C4⋊D4), C2.22(C207D4), C10.23(C8⋊C22), (C2×C20).776C23, (C2×D20).23C22, C22.135(C2×D20), (C22×C4).143D10, (C22×C10).104D4, C4⋊Dic5.27C22, (C22×C20).305C22, C4.114(C2×C5⋊D4), (C2×C10).166(C2×D4), (C2×C4).725(C22×D5), SmallGroup(320,762)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C403D4
C1C5C10C2×C10C2×C20C2×D20C2×D40 — C403D4
C5C10C2×C20 — C403D4
C1C22C22×C4C2×M4(2)

Generators and relations for C403D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >

Subgroups: 694 in 130 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], C22×C4, C2×D4 [×4], Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×3], D4⋊C4 [×2], C4.Q8, C4⋊D4 [×2], C2×M4(2), C2×D8, C40 [×2], C40, D20 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, C82D4, D40 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C2×C40 [×2], C5×M4(2) [×2], C2×D20 [×2], C2×C5⋊D4 [×2], C22×C20, C406C4, D205C4 [×2], C2×D40, C207D4 [×2], C10×M4(2), C403D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C82D4, C2×D20, C4○D20, C2×C5⋊D4, C8⋊D10 [×2], C207D4, C403D4

Smallest permutation representation of C403D4
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)
(1 128 102 46)(2 147 103 65)(3 126 104 44)(4 145 105 63)(5 124 106 42)(6 143 107 61)(7 122 108 80)(8 141 109 59)(9 160 110 78)(10 139 111 57)(11 158 112 76)(12 137 113 55)(13 156 114 74)(14 135 115 53)(15 154 116 72)(16 133 117 51)(17 152 118 70)(18 131 119 49)(19 150 120 68)(20 129 81 47)(21 148 82 66)(22 127 83 45)(23 146 84 64)(24 125 85 43)(25 144 86 62)(26 123 87 41)(27 142 88 60)(28 121 89 79)(29 140 90 58)(30 159 91 77)(31 138 92 56)(32 157 93 75)(33 136 94 54)(34 155 95 73)(35 134 96 52)(36 153 97 71)(37 132 98 50)(38 151 99 69)(39 130 100 48)(40 149 101 67)
(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 133)(42 132)(43 131)(44 130)(45 129)(46 128)(47 127)(48 126)(49 125)(50 124)(51 123)(52 122)(53 121)(54 160)(55 159)(56 158)(57 157)(58 156)(59 155)(60 154)(61 153)(62 152)(63 151)(64 150)(65 149)(66 148)(67 147)(68 146)(69 145)(70 144)(71 143)(72 142)(73 141)(74 140)(75 139)(76 138)(77 137)(78 136)(79 135)(80 134)(81 83)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444455888810···101010101020···202020202040···40
size11114404022440402244442···244442···244444···4

56 irreducible representations

dim1111112222222222244
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10C5⋊D4D20D20C4○D20C8⋊C22C8⋊D10
kernelC403D4C406C4D205C4C2×D40C207D4C10×M4(2)C40C2×C20C22×C10C2×M4(2)C20C2×C8C22×C4C8C2×C4C23C4C10C2
# reps1121212112242844828

Matrix representation of C403D4 in GL6(𝔽41)

090000
900000
003329813
00122284
003328812
0013372939
,
0400000
100000
0000400
0000351
001000
0064000
,
100000
0400000
001000
0064000
0000400
0000351

G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,33,12,33,13,0,0,29,2,28,37,0,0,8,28,8,29,0,0,13,4,12,39],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,40,35,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1] >;

C403D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_3D_4
% in TeX

G:=Group("C40:3D4");
// GroupNames label

G:=SmallGroup(320,762);
// by ID

G=gap.SmallGroup(320,762);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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