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G = C408D4order 320 = 26·5

8th semidirect product of C40 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C408D4, C57(C8⋊D4), C85(C5⋊D4), (C2×SD16)⋊1D5, C405C427C2, (C2×C8).90D10, D103Q86C2, (C10×SD16)⋊3C2, (C2×D4).75D10, C20.178(C2×D4), (C2×Q8).56D10, Q8⋊Dic531C2, D4⋊Dic536C2, C202D4.10C2, (C2×Dic5).81D4, (C22×D5).47D4, C22.271(D4×D5), C20.102(C4○D4), C4.33(D42D5), C2.30(D40⋊C2), C2.20(C202D4), C10.80(C8⋊C22), (C2×C20).451C23, (C2×C40).115C22, (Q8×C10).80C22, C10.117(C4⋊D4), C2.31(SD16⋊D5), (D4×C10).100C22, C10.51(C8.C22), C4⋊Dic5.178C22, (C2×C8⋊D5)⋊3C2, C4.83(C2×C5⋊D4), (C2×C4×D5).55C22, (C2×C10).363(C2×D4), (C2×C4).540(C22×D5), (C2×C52C8).160C22, SmallGroup(320,801)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C408D4
C1C5C10C2×C10C2×C20C2×C4×D5C2×C8⋊D5 — C408D4
C5C10C2×C20 — C408D4
C1C22C2×C4C2×SD16

Generators and relations for C408D4
 G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a29, cbc=b-1 >

Subgroups: 486 in 120 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, C8⋊D4, C8⋊D5, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C40, C5×SD16, C2×C4×D5, C2×C5⋊D4, D4×C10, Q8×C10, C405C4, D4⋊Dic5, Q8⋊Dic5, C2×C8⋊D5, C202D4, D103Q8, C10×SD16, C408D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C8⋊D4, D4×D5, D42D5, C2×C5⋊D4, D40⋊C2, SD16⋊D5, C202D4, C408D4

Smallest permutation representation of C408D4
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 95 64 142)(2 94 65 141)(3 93 66 140)(4 92 67 139)(5 91 68 138)(6 90 69 137)(7 89 70 136)(8 88 71 135)(9 87 72 134)(10 86 73 133)(11 85 74 132)(12 84 75 131)(13 83 76 130)(14 82 77 129)(15 81 78 128)(16 120 79 127)(17 119 80 126)(18 118 41 125)(19 117 42 124)(20 116 43 123)(21 115 44 122)(22 114 45 121)(23 113 46 160)(24 112 47 159)(25 111 48 158)(26 110 49 157)(27 109 50 156)(28 108 51 155)(29 107 52 154)(30 106 53 153)(31 105 54 152)(32 104 55 151)(33 103 56 150)(34 102 57 149)(35 101 58 148)(36 100 59 147)(37 99 60 146)(38 98 61 145)(39 97 62 144)(40 96 63 143)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 77)(42 66)(43 55)(45 73)(46 62)(47 51)(48 80)(49 69)(50 58)(52 76)(53 65)(56 72)(57 61)(59 79)(60 68)(63 75)(67 71)(70 78)(81 136)(82 125)(83 154)(84 143)(85 132)(86 121)(87 150)(88 139)(89 128)(90 157)(91 146)(92 135)(93 124)(94 153)(95 142)(96 131)(97 160)(98 149)(99 138)(100 127)(101 156)(102 145)(103 134)(104 123)(105 152)(106 141)(107 130)(108 159)(109 148)(110 137)(111 126)(112 155)(113 144)(114 133)(115 122)(116 151)(117 140)(118 129)(119 158)(120 147)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444455888810···1010101010202020202020202040···40
size1111820228204040224420202···28888444488884···4

44 irreducible representations

dim11111111222222222444444
type++++++++++++++++--++-
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4C8⋊C22C8.C22D42D5D4×D5D40⋊C2SD16⋊D5
kernelC408D4C405C4D4⋊Dic5Q8⋊Dic5C2×C8⋊D5C202D4D103Q8C10×SD16C40C2×Dic5C22×D5C2×SD16C20C2×C8C2×D4C2×Q8C8C10C10C4C22C2C2
# reps11111111211222228112244

Matrix representation of C408D4 in GL6(𝔽41)

100000
010000
0024281713
0013282813
0024282428
0013281328
,
4020000
4010000
001491323
0011271928
0013232732
0019283014
,
100000
1400000
001000
0064000
000010
0000640

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,13,24,13,0,0,28,28,28,28,0,0,17,28,24,13,0,0,13,13,28,28],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,14,11,13,19,0,0,9,27,23,28,0,0,13,19,27,30,0,0,23,28,32,14],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;

C408D4 in GAP, Magma, Sage, TeX

C_{40}\rtimes_8D_4
% in TeX

G:=Group("C40:8D4");
// GroupNames label

G:=SmallGroup(320,801);
// by ID

G=gap.SmallGroup(320,801);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,555,438,102,12550]);
// Polycyclic

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

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