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G = C5×C424S3order 480 = 25·3·5

Direct product of C5 and C424S3

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

Aliases: C5×C424S3, D121C20, Dic61C20, C60.190D4, C20.69D12, C1517C4≀C2, (C4×C12)⋊6C10, (C4×C20)⋊12S3, (C4×C60)⋊18C2, C4.6(S3×C20), C424(C5×S3), (C5×D12)⋊13C4, C20.95(C4×S3), C4.17(C5×D12), C12.33(C5×D4), C12.16(C2×C20), C60.203(C2×C4), C4○D12.1C10, (C5×Dic6)⋊13C4, (C2×C30).170D4, (C2×C20).419D6, C4.Dic31C10, C10.48(D6⋊C4), C30.90(C22⋊C4), (C2×C60).534C22, C31(C5×C4≀C2), C2.3(C5×D6⋊C4), (C2×C6).26(C5×D4), C6.1(C5×C22⋊C4), (C2×C4).69(S3×C10), (C5×C4○D12).7C2, C22.7(C5×C3⋊D4), (C5×C4.Dic3)⋊13C2, (C2×C12).101(C2×C10), (C2×C10).60(C3⋊D4), SmallGroup(480,124)

Series: Derived Chief Lower central Upper central

C1C12 — C5×C424S3
C1C3C6C12C2×C12C2×C60C5×C4○D12 — C5×C424S3
C3C6C12 — C5×C424S3
C1C20C2×C20C4×C20

Generators and relations for C5×C424S3
 G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, ebe=bc=cb, bd=db, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 212 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C10, C10 [×2], Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C5×S3, C30, C30, C4≀C2, C40, C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8, C4.Dic3, C4×C12, C4○D12, C5×Dic3, C60 [×2], C60 [×2], S3×C10, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, C424S3, C5×C3⋊C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, C2×C60, C5×C4≀C2, C5×C4.Dic3, C4×C60, C5×C4○D12, C5×C424S3
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4≀C2, C2×C20, C5×D4 [×2], D6⋊C4, S3×C10, C5×C22⋊C4, C424S3, S3×C20, C5×D12, C5×C3⋊D4, C5×C4≀C2, C5×D6⋊C4, C5×C424S3

Smallest permutation representation of C5×C424S3
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 58)(2 59)(3 60)(4 56)(5 57)(6 43)(7 44)(8 45)(9 41)(10 42)(11 81 48 111)(12 82 49 112)(13 83 50 113)(14 84 46 114)(15 85 47 115)(16 66)(17 67)(18 68)(19 69)(20 70)(21 120 105 100)(22 116 101 96)(23 117 102 97)(24 118 103 98)(25 119 104 99)(26 89)(27 90)(28 86)(29 87)(30 88)(31 51)(32 52)(33 53)(34 54)(35 55)(36 73 79 110)(37 74 80 106)(38 75 76 107)(39 71 77 108)(40 72 78 109)(61 91)(62 92)(63 93)(64 94)(65 95)
(1 16 58 66)(2 17 59 67)(3 18 60 68)(4 19 56 69)(5 20 57 70)(6 93 43 63)(7 94 44 64)(8 95 45 65)(9 91 41 61)(10 92 42 62)(11 81 48 111)(12 82 49 112)(13 83 50 113)(14 84 46 114)(15 85 47 115)(21 120 105 100)(22 116 101 96)(23 117 102 97)(24 118 103 98)(25 119 104 99)(26 31 89 51)(27 32 90 52)(28 33 86 53)(29 34 87 54)(30 35 88 55)(36 73 79 110)(37 74 80 106)(38 75 76 107)(39 71 77 108)(40 72 78 109)
(1 8 90)(2 9 86)(3 10 87)(4 6 88)(5 7 89)(11 24 73)(12 25 74)(13 21 75)(14 22 71)(15 23 72)(16 95 52)(17 91 53)(18 92 54)(19 93 55)(20 94 51)(26 57 44)(27 58 45)(28 59 41)(29 60 42)(30 56 43)(31 70 64)(32 66 65)(33 67 61)(34 68 62)(35 69 63)(36 111 98)(37 112 99)(38 113 100)(39 114 96)(40 115 97)(46 101 108)(47 102 109)(48 103 110)(49 104 106)(50 105 107)(76 83 120)(77 84 116)(78 85 117)(79 81 118)(80 82 119)
(1 98)(2 99)(3 100)(4 96)(5 97)(6 114)(7 115)(8 111)(9 112)(10 113)(11 65)(12 61)(13 62)(14 63)(15 64)(16 103)(17 104)(18 105)(19 101)(20 102)(21 68)(22 69)(23 70)(24 66)(25 67)(26 78)(27 79)(28 80)(29 76)(30 77)(31 72)(32 73)(33 74)(34 75)(35 71)(36 90)(37 86)(38 87)(39 88)(40 89)(41 82)(42 83)(43 84)(44 85)(45 81)(46 93)(47 94)(48 95)(49 91)(50 92)(51 109)(52 110)(53 106)(54 107)(55 108)(56 116)(57 117)(58 118)(59 119)(60 120)

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

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

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

150 conjugacy classes

class 1 2A2B2C 3 4A4B4C···4G4H5A5B5C5D6A6B6C8A8B10A10B10C10D10E10F10G10H10I10J10K10L12A···12L15A15B15C15D20A···20H20I···20AB20AC20AD20AE20AF30A···30L40A···40H60A···60AV
order12223444···4455556668810101010101010101010101012···121515151520···2020···202020202030···3040···4060···60
size112122112···2121111222121211112222121212122···222221···12···2121212122···212···122···2

150 irreducible representations

dim111111111111222222222222222222
type+++++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20S3D4D4D6C4×S3D12C3⋊D4C5×S3C4≀C2C5×D4C5×D4S3×C10C424S3S3×C20C5×D12C5×C3⋊D4C5×C4≀C2C5×C424S3
kernelC5×C424S3C5×C4.Dic3C4×C60C5×C4○D12C5×Dic6C5×D12C424S3C4.Dic3C4×C12C4○D12Dic6D12C4×C20C60C2×C30C2×C20C20C20C2×C10C42C15C12C2×C6C2×C4C5C4C4C22C3C1
# reps11112244448811112224444488881632

Matrix representation of C5×C424S3 in GL2(𝔽241) generated by

2050
0205
,
2400
8164
,
1770
20464
,
150
169225
,
3172
56238
G:=sub<GL(2,GF(241))| [205,0,0,205],[240,81,0,64],[177,204,0,64],[15,169,0,225],[3,56,172,238] >;

C5×C424S3 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes_4S_3
% in TeX

G:=Group("C5xC4^2:4S3");
// GroupNames label

G:=SmallGroup(480,124);
// by ID

G=gap.SmallGroup(480,124);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,2803,4204,102,15686]);
// Polycyclic

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

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×
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