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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 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8, C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5, C10 [×3], C10, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C52C8, C40 [×2], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C8⋊D4, C8⋊D5 [×2], C2×C52C8, C10.D4, C4⋊Dic5 [×2], D10⋊C4, C23.D5, C2×C40, C5×SD16 [×2], 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 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, C5⋊D4 [×2], 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)
(1 146 102 66)(2 145 103 65)(3 144 104 64)(4 143 105 63)(5 142 106 62)(6 141 107 61)(7 140 108 60)(8 139 109 59)(9 138 110 58)(10 137 111 57)(11 136 112 56)(12 135 113 55)(13 134 114 54)(14 133 115 53)(15 132 116 52)(16 131 117 51)(17 130 118 50)(18 129 119 49)(19 128 120 48)(20 127 81 47)(21 126 82 46)(22 125 83 45)(23 124 84 44)(24 123 85 43)(25 122 86 42)(26 121 87 41)(27 160 88 80)(28 159 89 79)(29 158 90 78)(30 157 91 77)(31 156 92 76)(32 155 93 75)(33 154 94 74)(34 153 95 73)(35 152 96 72)(36 151 97 71)(37 150 98 70)(38 149 99 69)(39 148 100 68)(40 147 101 67)
(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)(41 141)(42 130)(43 159)(44 148)(45 137)(46 126)(47 155)(48 144)(49 133)(50 122)(51 151)(52 140)(53 129)(54 158)(55 147)(56 136)(57 125)(58 154)(59 143)(60 132)(61 121)(62 150)(63 139)(64 128)(65 157)(66 146)(67 135)(68 124)(69 153)(70 142)(71 131)(72 160)(73 149)(74 138)(75 127)(76 156)(77 145)(78 134)(79 123)(80 152)(81 93)(83 111)(84 100)(85 89)(86 118)(87 107)(88 96)(90 114)(91 103)(94 110)(95 99)(97 117)(98 106)(101 113)(104 120)(105 109)(108 116)(115 119)

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

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

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

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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