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G = C5×D12⋊C4order 480 = 25·3·5

Direct product of C5 and D12⋊C4

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

Aliases: C5×D12⋊C4, D124C20, Dic64C20, C60.239D4, C1520C4≀C2, C4.3(S3×C20), (C5×D12)⋊16C4, C20.78(C4×S3), C12.6(C2×C20), (C2×C30).88D4, C12.54(C5×D4), C60.176(C2×C4), C4○D12.2C10, (C5×Dic6)⋊16C4, (C4×Dic3)⋊1C10, (C2×C20).349D6, (C2×C10).25D12, M4(2)⋊4(C5×S3), (C5×M4(2))⋊8S3, C22.3(C5×D12), C10.57(D6⋊C4), (Dic3×C20)⋊13C2, (C3×M4(2))⋊8C10, C20.122(C3⋊D4), C30.99(C22⋊C4), (C15×M4(2))⋊18C2, (C2×C60).344C22, C32(C5×C4≀C2), (C2×C6).1(C5×D4), C2.11(C5×D6⋊C4), C4.29(C5×C3⋊D4), (C2×C4).37(S3×C10), (C5×C4○D12).8C2, C6.10(C5×C22⋊C4), (C2×C12).14(C2×C10), SmallGroup(480,144)

Series: Derived Chief Lower central Upper central

C1C12 — C5×D12⋊C4
C1C3C6C12C2×C12C2×C60C5×C4○D12 — C5×D12⋊C4
C3C6C12 — C5×D12⋊C4
C1C20C2×C20C5×M4(2)

Generators and relations for C5×D12⋊C4
 G = < a,b,c,d | a5=b12=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b7c >

Subgroups: 228 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 [×3], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C4≀C2, C40, C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8, C4×Dic3, C3×M4(2), C4○D12, C5×Dic3 [×3], C60 [×2], S3×C10, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, D12⋊C4, C120, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, C5×C4≀C2, Dic3×C20, C15×M4(2), C5×C4○D12, C5×D12⋊C4
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, D12⋊C4, S3×C20, C5×D12, C5×C3⋊D4, C5×C4≀C2, C5×D6⋊C4, C5×D12⋊C4

Smallest permutation representation of C5×D12⋊C4
On 120 points
Generators in S120
(1 59 38 32 14)(2 60 39 33 15)(3 49 40 34 16)(4 50 41 35 17)(5 51 42 36 18)(6 52 43 25 19)(7 53 44 26 20)(8 54 45 27 21)(9 55 46 28 22)(10 56 47 29 23)(11 57 48 30 24)(12 58 37 31 13)(61 118 106 95 78)(62 119 107 96 79)(63 120 108 85 80)(64 109 97 86 81)(65 110 98 87 82)(66 111 99 88 83)(67 112 100 89 84)(68 113 101 90 73)(69 114 102 91 74)(70 115 103 92 75)(71 116 104 93 76)(72 117 105 94 77)
(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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 72)(13 77)(14 76)(15 75)(16 74)(17 73)(18 84)(19 83)(20 82)(21 81)(22 80)(23 79)(24 78)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 105)(38 104)(39 103)(40 102)(41 101)(42 100)(43 99)(44 98)(45 97)(46 108)(47 107)(48 106)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)
(1 4 7 10)(2 9 8 3)(5 12 11 6)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 36 31 30)(26 29 32 35)(27 34 33 28)(37 48 43 42)(38 41 44 47)(39 46 45 40)(49 60 55 54)(50 53 56 59)(51 58 57 52)(61 65)(62 70)(64 68)(67 71)(73 81)(75 79)(76 84)(78 82)(86 90)(87 95)(89 93)(92 96)(97 101)(98 106)(100 104)(103 107)(109 113)(110 118)(112 116)(115 119)

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B5C5D6A6B8A8B10A10B10C10D10E10F10G10H10I10J10K10L12A12B12C15A15B15C15D20A···20H20I20J20K20L20M···20AB20AC20AD20AE20AF24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1222344444444555566881010101010101010101010101212121515151520···202020202020···202020202024242424303030303030303040···4060···6060606060120···120
size11212211266661211112444111122221212121222422221···122226···6121212124444222244444···42···244444···4

120 irreducible representations

dim111111111111222222222222222244
type+++++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20S3D4D4D6C4×S3C3⋊D4D12C5×S3C4≀C2C5×D4C5×D4S3×C10S3×C20C5×C3⋊D4C5×D12C5×C4≀C2D12⋊C4C5×D12⋊C4
kernelC5×D12⋊C4Dic3×C20C15×M4(2)C5×C4○D12C5×Dic6C5×D12D12⋊C4C4×Dic3C3×M4(2)C4○D12Dic6D12C5×M4(2)C60C2×C30C2×C20C20C20C2×C10M4(2)C15C12C2×C6C2×C4C4C4C22C3C5C1
# reps1111224444881111222444448881628

Matrix representation of C5×D12⋊C4 in GL4(𝔽241) generated by

91000
09100
002050
000205
,
024000
1100
00640
000177
,
4314200
9919800
0001
0010
,
240000
1100
001770
0001
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,205,0,0,0,0,205],[0,1,0,0,240,1,0,0,0,0,64,0,0,0,0,177],[43,99,0,0,142,198,0,0,0,0,0,1,0,0,1,0],[240,1,0,0,0,1,0,0,0,0,177,0,0,0,0,1] >;

C5×D12⋊C4 in GAP, Magma, Sage, TeX

C_5\times D_{12}\rtimes C_4
% in TeX

G:=Group("C5xD12:C4");
// GroupNames label

G:=SmallGroup(480,144);
// by ID

G=gap.SmallGroup(480,144);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,136,4204,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^7*c>;
// generators/relations

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