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G = D4015C4order 320 = 26·5

9th semidirect product of D40 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4015C4, C84(C4×D5), C4018(C2×C4), C4.Q85D5, C408C43C2, D2016(C2×C4), C54(D8⋊C4), C10.81(C4×D4), (C2×C8).63D10, D208C46C2, C4⋊C4.163D10, (C2×D40).13C2, D206C418C2, C22.86(D4×D5), C20.34(C4○D4), C2.6(D40⋊C2), C4.6(Q82D5), C10.72(C8⋊C22), C20.104(C22×C4), (C2×C20).285C23, (C2×C40).112C22, (C2×Dic5).221D4, (C2×D20).81C22, C2.11(D208C4), (C4×Dic5).37C22, C4.42(C2×C4×D5), (C5×C4.Q8)⋊5C2, (C2×C10).290(C2×D4), (C5×C4⋊C4).78C22, (C2×C52C8).62C22, (C2×C4).388(C22×D5), SmallGroup(320,496)

Series: Derived Chief Lower central Upper central

C1C20 — D4015C4
C1C5C10C2×C10C2×C20C2×D20C2×D40 — D4015C4
C5C10C20 — D4015C4
C1C22C2×C4C4.Q8

Generators and relations for D4015C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a10b >

Subgroups: 646 in 132 conjugacy classes, 49 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C8, C2×C4, C2×C4 [×8], D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×8], C2×C10, C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, C52C8, C40 [×2], C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], D8⋊C4, D40 [×4], C2×C52C8, C4×Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5 [×2], C2×D20 [×2], D206C4 [×2], C408C4, C5×C4.Q8, D208C4 [×2], C2×D40, D4015C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22 [×2], C4×D5 [×2], C22×D5, D8⋊C4, C2×C4×D5, D4×D5, Q82D5, D208C4, D40⋊C2 [×2], D4015C4

Smallest permutation representation of D4015C4
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(70 80)(71 79)(72 78)(73 77)(74 76)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 151)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)(152 160)(153 159)(154 158)(155 157)
(1 154 48 91)(2 125 49 102)(3 136 50 113)(4 147 51 84)(5 158 52 95)(6 129 53 106)(7 140 54 117)(8 151 55 88)(9 122 56 99)(10 133 57 110)(11 144 58 81)(12 155 59 92)(13 126 60 103)(14 137 61 114)(15 148 62 85)(16 159 63 96)(17 130 64 107)(18 141 65 118)(19 152 66 89)(20 123 67 100)(21 134 68 111)(22 145 69 82)(23 156 70 93)(24 127 71 104)(25 138 72 115)(26 149 73 86)(27 160 74 97)(28 131 75 108)(29 142 76 119)(30 153 77 90)(31 124 78 101)(32 135 79 112)(33 146 80 83)(34 157 41 94)(35 128 42 105)(36 139 43 116)(37 150 44 87)(38 121 45 98)(39 132 46 109)(40 143 47 120)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222222444444444455888810···102020202020···2040···40
size11112020202022444410101010224420202···244448···84···4

50 irreducible representations

dim11111112222224444
type++++++++++++++
imageC1C2C2C2C2C2C4D4D5C4○D4D10D10C4×D5C8⋊C22Q82D5D4×D5D40⋊C2
kernelD4015C4D206C4C408C4C5×C4.Q8D208C4C2×D40D40C2×Dic5C4.Q8C20C4⋊C4C2×C8C8C10C4C22C2
# reps12112182224282228

Matrix representation of D4015C4 in GL6(𝔽41)

30360000
8110000
0013281328
0013171317
0028131328
0028241317
,
4000000
2910000
000066
0000135
006600
0013500
,
24370000
11170000
0038241817
0017182438
001817317
0024382423

G:=sub<GL(6,GF(41))| [30,8,0,0,0,0,36,11,0,0,0,0,0,0,13,13,28,28,0,0,28,17,13,24,0,0,13,13,13,13,0,0,28,17,28,17],[40,29,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,6,1,0,0,0,0,6,35,0,0],[24,11,0,0,0,0,37,17,0,0,0,0,0,0,38,17,18,24,0,0,24,18,17,38,0,0,18,24,3,24,0,0,17,38,17,23] >;

D4015C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_{15}C_4
% in TeX

G:=Group("D40:15C4");
// GroupNames label

G:=SmallGroup(320,496);
// by ID

G=gap.SmallGroup(320,496);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,219,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^10*b>;
// generators/relations

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