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G = C409D4order 320 = 26·5

9th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C409D4, C52C810D4, C86(C5⋊D4), C55(C83D4), C408C46C2, C4.26(D4×D5), (C2×D40)⋊25C2, C20⋊D46C2, (C2×SD16)⋊2D5, (C2×C8).91D10, (C10×SD16)⋊4C2, (C2×D4).77D10, C20.179(C2×D4), (C2×Q8).58D10, C20.23D45C2, (C2×Dic5).82D4, C22.273(D4×D5), C2.23(C20⋊D4), C10.32(C41D4), C2.31(D40⋊C2), C10.81(C8⋊C22), (C2×C40).116C22, (C2×C20).453C23, (Q8×C10).82C22, (C2×D20).127C22, (D4×C10).102C22, (C4×Dic5).58C22, (C2×D4⋊D5)⋊21C2, (C2×Q8⋊D5)⋊19C2, C4.10(C2×C5⋊D4), (C2×C10).365(C2×D4), (C2×C4).542(C22×D5), (C2×C52C8).161C22, SmallGroup(320,803)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C409D4
C1C5C10C2×C10C2×C20C2×D20C2×D40 — C409D4
C5C10C2×C20 — C409D4
C1C22C2×C4C2×SD16

Generators and relations for C409D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=a-1, cbc=b-1 >

Subgroups: 750 in 144 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×3], D4 [×10], Q8 [×2], C23 [×3], D5 [×2], C10, C10 [×2], C10, C42, C22⋊C4 [×2], C2×C8, C2×C8, D8 [×4], SD16 [×4], C2×D4, C2×D4 [×4], C2×Q8, Dic5 [×2], C20 [×2], C20, D10 [×6], C2×C10, C2×C10 [×3], C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16, C2×SD16, C52C8 [×2], C40 [×2], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10, C83D4, D40 [×2], C2×C52C8, C4×Dic5, D10⋊C4 [×2], D4⋊D5 [×2], Q8⋊D5 [×2], C2×C40, C5×SD16 [×2], C2×D20 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C408C4, C2×D40, C2×D4⋊D5, C20⋊D4, C2×Q8⋊D5, C20.23D4, C10×SD16, C409D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C8⋊C22 [×2], C5⋊D4 [×2], C22×D5, C83D4, D4×D5 [×2], C2×C5⋊D4, D40⋊C2 [×2], C20⋊D4, C409D4

Smallest permutation representation of C409D4
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)
(1 157 71 106)(2 146 72 95)(3 135 73 84)(4 124 74 113)(5 153 75 102)(6 142 76 91)(7 131 77 120)(8 160 78 109)(9 149 79 98)(10 138 80 87)(11 127 41 116)(12 156 42 105)(13 145 43 94)(14 134 44 83)(15 123 45 112)(16 152 46 101)(17 141 47 90)(18 130 48 119)(19 159 49 108)(20 148 50 97)(21 137 51 86)(22 126 52 115)(23 155 53 104)(24 144 54 93)(25 133 55 82)(26 122 56 111)(27 151 57 100)(28 140 58 89)(29 129 59 118)(30 158 60 107)(31 147 61 96)(32 136 62 85)(33 125 63 114)(34 154 64 103)(35 143 65 92)(36 132 66 81)(37 121 67 110)(38 150 68 99)(39 139 69 88)(40 128 70 117)
(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 61)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(81 142)(82 141)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 122)(102 121)(103 160)(104 159)(105 158)(106 157)(107 156)(108 155)(109 154)(110 153)(111 152)(112 151)(113 150)(114 149)(115 148)(116 147)(117 146)(118 145)(119 144)(120 143)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444455888810···1010101010202020202020202040···40
size1111840402282020224420202···28888444488884···4

44 irreducible representations

dim11111111222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5D10D10D10C5⋊D4C8⋊C22D4×D5D4×D5D40⋊C2
kernelC409D4C408C4C2×D40C2×D4⋊D5C20⋊D4C2×Q8⋊D5C20.23D4C10×SD16C52C8C40C2×Dic5C2×SD16C2×C8C2×D4C2×Q8C8C10C4C22C2
# reps11111111222222282228

Matrix representation of C409D4 in GL8(𝔽41)

281925140000
221927270000
162713220000
141419220000
0000324324
0000374374
0000937324
0000437374
,
004000000
003510000
10000000
640000000
00001401516
000025273426
00002625140
00007152527
,
10000000
640000000
004000000
003510000
00001000
0000344000
000000400
00000071

G:=sub<GL(8,GF(41))| [28,22,16,14,0,0,0,0,19,19,27,14,0,0,0,0,25,27,13,19,0,0,0,0,14,27,22,22,0,0,0,0,0,0,0,0,32,37,9,4,0,0,0,0,4,4,37,37,0,0,0,0,32,37,32,37,0,0,0,0,4,4,4,4],[0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,14,25,26,7,0,0,0,0,0,27,25,15,0,0,0,0,15,34,14,25,0,0,0,0,16,26,0,27],[1,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,1] >;

C409D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_9D_4
% in TeX

G:=Group("C40:9D4");
// GroupNames label

G:=SmallGroup(320,803);
// by ID

G=gap.SmallGroup(320,803);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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