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G = C5×Q82S3order 240 = 24·3·5

Direct product of C5 and Q82S3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×Q82S3, C30.49D4, C20.38D6, C1515SD16, D12.2C10, C60.45C22, C3⋊C83C10, Q82(C5×S3), (C5×Q8)⋊6S3, C6.9(C5×D4), C33(C5×SD16), C4.3(S3×C10), (C3×Q8)⋊1C10, (Q8×C15)⋊7C2, C12.3(C2×C10), (C5×D12).4C2, C10.25(C3⋊D4), (C5×C3⋊C8)⋊10C2, C2.6(C5×C3⋊D4), SmallGroup(240,62)

Series: Derived Chief Lower central Upper central

C1C12 — C5×Q82S3
C1C3C6C12C60C5×D12 — C5×Q82S3
C3C6C12 — C5×Q82S3
C1C10C20C5×Q8

Generators and relations for C5×Q82S3
 G = < a,b,c,d,e | a5=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

12C2
2C4
6C22
4S3
12C10
3D4
3C8
2C12
2D6
2C20
6C2×C10
4C5×S3
3SD16
3C5×D4
3C40
2C60
2S3×C10
3C5×SD16

Smallest permutation representation of C5×Q82S3
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 26 58 66)(2 27 59 67)(3 28 60 68)(4 29 56 69)(5 30 57 70)(6 20 90 52)(7 16 86 53)(8 17 87 54)(9 18 88 55)(10 19 89 51)(11 98 48 111)(12 99 49 112)(13 100 50 113)(14 96 46 114)(15 97 47 115)(21 75 93 107)(22 71 94 108)(23 72 95 109)(24 73 91 110)(25 74 92 106)(31 76 43 63)(32 77 44 64)(33 78 45 65)(34 79 41 61)(35 80 42 62)(36 118 103 81)(37 119 104 82)(38 120 105 83)(39 116 101 84)(40 117 102 85)
(1 118 58 81)(2 119 59 82)(3 120 60 83)(4 116 56 84)(5 117 57 85)(6 24 90 91)(7 25 86 92)(8 21 87 93)(9 22 88 94)(10 23 89 95)(11 65 48 78)(12 61 49 79)(13 62 50 80)(14 63 46 76)(15 64 47 77)(16 106 53 74)(17 107 54 75)(18 108 55 71)(19 109 51 72)(20 110 52 73)(26 36 66 103)(27 37 67 104)(28 38 68 105)(29 39 69 101)(30 40 70 102)(31 96 43 114)(32 97 44 115)(33 98 45 111)(34 99 41 112)(35 100 42 113)
(1 33 90)(2 34 86)(3 35 87)(4 31 88)(5 32 89)(6 58 45)(7 59 41)(8 60 42)(9 56 43)(10 57 44)(11 73 36)(12 74 37)(13 75 38)(14 71 39)(15 72 40)(16 67 61)(17 68 62)(18 69 63)(19 70 64)(20 66 65)(21 83 113)(22 84 114)(23 85 115)(24 81 111)(25 82 112)(26 78 52)(27 79 53)(28 80 54)(29 76 55)(30 77 51)(46 108 101)(47 109 102)(48 110 103)(49 106 104)(50 107 105)(91 118 98)(92 119 99)(93 120 100)(94 116 96)(95 117 97)
(6 45)(7 41)(8 42)(9 43)(10 44)(11 24)(12 25)(13 21)(14 22)(15 23)(16 79)(17 80)(18 76)(19 77)(20 78)(26 66)(27 67)(28 68)(29 69)(30 70)(31 88)(32 89)(33 90)(34 86)(35 87)(36 81)(37 82)(38 83)(39 84)(40 85)(46 94)(47 95)(48 91)(49 92)(50 93)(51 64)(52 65)(53 61)(54 62)(55 63)(71 114)(72 115)(73 111)(74 112)(75 113)(96 108)(97 109)(98 110)(99 106)(100 107)(101 116)(102 117)(103 118)(104 119)(105 120)

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

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

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

C5×Q82S3 is a maximal subgroup of
D20⋊D6  D15⋊SD16  D60⋊C22  D12.27D10  D20.14D6  D12.D10  D30.44D4  C5×S3×SD16

60 conjugacy classes

class 1 2A2B 3 4A4B5A5B5C5D 6 8A8B10A10B10C10D10E10F10G10H12A12B12C15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D40A···40H60A···60L
order122344555568810101010101010101212121515151520202020202020203030303040···4060···60
size1112224111126611111212121244422222222444422226···64···4

60 irreducible representations

dim11111111222222222244
type++++++++
imageC1C2C2C2C5C10C10C10S3D4D6SD16C3⋊D4C5×S3C5×D4S3×C10C5×SD16C5×C3⋊D4Q82S3C5×Q82S3
kernelC5×Q82S3C5×C3⋊C8C5×D12Q8×C15Q82S3C3⋊C8D12C3×Q8C5×Q8C30C20C15C10Q8C6C4C3C2C5C1
# reps11114444111224448814

Matrix representation of C5×Q82S3 in GL4(𝔽241) generated by

91000
09100
0010
0001
,
1000
0100
00240239
0011
,
1000
0100
00038
00190
,
240100
240000
0010
0001
,
0100
1000
0010
00240240
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,240,1,0,0,239,1],[1,0,0,0,0,1,0,0,0,0,0,19,0,0,38,0],[240,240,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,240,0,0,0,240] >;

C5×Q82S3 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_2S_3
% in TeX

G:=Group("C5xQ8:2S3");
// GroupNames label

G:=SmallGroup(240,62);
// by ID

G=gap.SmallGroup(240,62);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,265,247,1443,729,69,5765]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of C5×Q82S3 in TeX

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