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G = C405D6order 480 = 25·3·5

5th semidirect product of C40 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C405D6, D404S3, D151D8, D201D6, D244D5, C245D10, D121D10, D30.21D4, C12014C22, C60.140C23, Dic15.26D4, C52(S3×D8), C32(D5×D8), C153(C2×D8), C813(S3×D5), (C3×D40)⋊7C2, (C8×D15)⋊6C2, (C5×D24)⋊6C2, C15⋊D89C2, C6.29(D4×D5), C20⋊D68C2, C30.10(C2×D4), C10.29(S3×D4), C153C836C22, C2.7(C20⋊D6), (C5×D12)⋊16C22, (C3×D20)⋊16C22, C20.69(C22×S3), C12.69(C22×D5), (C4×D15).54C22, C4.113(C2×S3×D5), SmallGroup(480,332)

Series: Derived Chief Lower central Upper central

C1C60 — C405D6
C1C5C15C30C60C3×D20C20⋊D6 — C405D6
C15C30C60 — C405D6
C1C2C4C8

Generators and relations for C405D6
 G = < a,b,c | a40=b6=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 1180 in 152 conjugacy classes, 40 normal (26 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×4], C6, C6 [×2], C8, C8, C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6 [×7], C2×C6 [×2], C15, C2×C8, D8 [×4], C2×D4 [×2], Dic5, C20, D10 [×7], C2×C10 [×2], C3⋊C8, C24, C4×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C2×D8, C52C8, C40, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C22×D5 [×2], S3×C8, D24, D4⋊S3 [×2], C3×D8, S3×D4 [×2], Dic15, C60, S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30, C8×D5, D40, D4⋊D5 [×2], C5×D8, D4×D5 [×2], S3×D8, C153C8, C120, C15⋊D4 [×2], C3×D20 [×2], C5×D12 [×2], C4×D15, C2×S3×D5 [×2], D5×D8, C15⋊D8 [×2], C3×D40, C5×D24, C8×D15, C20⋊D6 [×2], C405D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], C22×S3, C2×D8, C22×D5, S3×D4, S3×D5, D4×D5, S3×D8, C2×S3×D5, D5×D8, C20⋊D6, C405D6

Smallest permutation representation of C405D6
On 120 points
Generators in S120
(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)
(1 46 109 16 71 84)(2 45 110 15 72 83)(3 44 111 14 73 82)(4 43 112 13 74 81)(5 42 113 12 75 120)(6 41 114 11 76 119)(7 80 115 10 77 118)(8 79 116 9 78 117)(17 70 85 40 47 108)(18 69 86 39 48 107)(19 68 87 38 49 106)(20 67 88 37 50 105)(21 66 89 36 51 104)(22 65 90 35 52 103)(23 64 91 34 53 102)(24 63 92 33 54 101)(25 62 93 32 55 100)(26 61 94 31 56 99)(27 60 95 30 57 98)(28 59 96 29 58 97)
(1 89)(2 98)(3 107)(4 116)(5 85)(6 94)(7 103)(8 112)(9 81)(10 90)(11 99)(12 108)(13 117)(14 86)(15 95)(16 104)(17 113)(18 82)(19 91)(20 100)(21 109)(22 118)(23 87)(24 96)(25 105)(26 114)(27 83)(28 92)(29 101)(30 110)(31 119)(32 88)(33 97)(34 106)(35 115)(36 84)(37 93)(38 102)(39 111)(40 120)(41 61)(42 70)(43 79)(44 48)(45 57)(46 66)(47 75)(49 53)(50 62)(51 71)(52 80)(54 58)(55 67)(56 76)(59 63)(60 72)(64 68)(65 77)(69 73)(74 78)

G:=sub<Sym(120)| (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), (1,46,109,16,71,84)(2,45,110,15,72,83)(3,44,111,14,73,82)(4,43,112,13,74,81)(5,42,113,12,75,120)(6,41,114,11,76,119)(7,80,115,10,77,118)(8,79,116,9,78,117)(17,70,85,40,47,108)(18,69,86,39,48,107)(19,68,87,38,49,106)(20,67,88,37,50,105)(21,66,89,36,51,104)(22,65,90,35,52,103)(23,64,91,34,53,102)(24,63,92,33,54,101)(25,62,93,32,55,100)(26,61,94,31,56,99)(27,60,95,30,57,98)(28,59,96,29,58,97), (1,89)(2,98)(3,107)(4,116)(5,85)(6,94)(7,103)(8,112)(9,81)(10,90)(11,99)(12,108)(13,117)(14,86)(15,95)(16,104)(17,113)(18,82)(19,91)(20,100)(21,109)(22,118)(23,87)(24,96)(25,105)(26,114)(27,83)(28,92)(29,101)(30,110)(31,119)(32,88)(33,97)(34,106)(35,115)(36,84)(37,93)(38,102)(39,111)(40,120)(41,61)(42,70)(43,79)(44,48)(45,57)(46,66)(47,75)(49,53)(50,62)(51,71)(52,80)(54,58)(55,67)(56,76)(59,63)(60,72)(64,68)(65,77)(69,73)(74,78)>;

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), (1,46,109,16,71,84)(2,45,110,15,72,83)(3,44,111,14,73,82)(4,43,112,13,74,81)(5,42,113,12,75,120)(6,41,114,11,76,119)(7,80,115,10,77,118)(8,79,116,9,78,117)(17,70,85,40,47,108)(18,69,86,39,48,107)(19,68,87,38,49,106)(20,67,88,37,50,105)(21,66,89,36,51,104)(22,65,90,35,52,103)(23,64,91,34,53,102)(24,63,92,33,54,101)(25,62,93,32,55,100)(26,61,94,31,56,99)(27,60,95,30,57,98)(28,59,96,29,58,97), (1,89)(2,98)(3,107)(4,116)(5,85)(6,94)(7,103)(8,112)(9,81)(10,90)(11,99)(12,108)(13,117)(14,86)(15,95)(16,104)(17,113)(18,82)(19,91)(20,100)(21,109)(22,118)(23,87)(24,96)(25,105)(26,114)(27,83)(28,92)(29,101)(30,110)(31,119)(32,88)(33,97)(34,106)(35,115)(36,84)(37,93)(38,102)(39,111)(40,120)(41,61)(42,70)(43,79)(44,48)(45,57)(46,66)(47,75)(49,53)(50,62)(51,71)(52,80)(54,58)(55,67)(56,76)(59,63)(60,72)(64,68)(65,77)(69,73)(74,78) );

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)], [(1,46,109,16,71,84),(2,45,110,15,72,83),(3,44,111,14,73,82),(4,43,112,13,74,81),(5,42,113,12,75,120),(6,41,114,11,76,119),(7,80,115,10,77,118),(8,79,116,9,78,117),(17,70,85,40,47,108),(18,69,86,39,48,107),(19,68,87,38,49,106),(20,67,88,37,50,105),(21,66,89,36,51,104),(22,65,90,35,52,103),(23,64,91,34,53,102),(24,63,92,33,54,101),(25,62,93,32,55,100),(26,61,94,31,56,99),(27,60,95,30,57,98),(28,59,96,29,58,97)], [(1,89),(2,98),(3,107),(4,116),(5,85),(6,94),(7,103),(8,112),(9,81),(10,90),(11,99),(12,108),(13,117),(14,86),(15,95),(16,104),(17,113),(18,82),(19,91),(20,100),(21,109),(22,118),(23,87),(24,96),(25,105),(26,114),(27,83),(28,92),(29,101),(30,110),(31,119),(32,88),(33,97),(34,106),(35,115),(36,84),(37,93),(38,102),(39,111),(40,120),(41,61),(42,70),(43,79),(44,48),(45,57),(46,66),(47,75),(49,53),(50,62),(51,71),(52,80),(54,58),(55,67),(56,76),(59,63),(60,72),(64,68),(65,77),(69,73),(74,78)])

51 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C8A8B8C8D10A10B10C10D10E10F 12 15A15B20A20B24A24B30A30B40A40B40C40D60A60B60C60D120A···120H
order122222223445566688881010101010101215152020242430304040404060606060120···120
size11121215152020223022240402230302224242424444444444444444444···4

51 irreducible representations

dim11111122222222244444444
type+++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D8D10D10S3×D4S3×D5D4×D5S3×D8C2×S3×D5D5×D8C20⋊D6C405D6
kernelC405D6C15⋊D8C3×D40C5×D24C8×D15C20⋊D6D40Dic15D30D24C40D20D15C24D12C10C8C6C5C4C3C2C1
# reps12111211121242412222448

Matrix representation of C405D6 in GL6(𝔽241)

01900000
52520000
0002200
002302200
00002400
00000240
,
52510000
1881890000
0002200
0011000
000001
00002401
,
1891900000
53520000
00240000
00024000
00001240
00000240

G:=sub<GL(6,GF(241))| [0,52,0,0,0,0,190,52,0,0,0,0,0,0,0,230,0,0,0,0,22,22,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[52,188,0,0,0,0,51,189,0,0,0,0,0,0,0,11,0,0,0,0,22,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[189,53,0,0,0,0,190,52,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;

C405D6 in GAP, Magma, Sage, TeX

C_{40}\rtimes_5D_6
% in TeX

G:=Group("C40:5D6");
// GroupNames label

G:=SmallGroup(480,332);
// by ID

G=gap.SmallGroup(480,332);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,142,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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