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G = C5×D6⋊C4order 240 = 24·3·5

Direct product of C5 and D6⋊C4

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

Aliases: C5×D6⋊C4, D6⋊C20, C30.46D4, C10.17D12, (C2×C20)⋊1S3, (C2×C60)⋊2C2, C6.6(C5×D4), (S3×C10)⋊5C4, (C2×C12)⋊1C10, C2.5(S3×C20), C6.4(C2×C20), C2.2(C5×D12), C10.26(C4×S3), C159(C22⋊C4), C30.49(C2×C4), (C2×C10).34D6, (C22×S3).C10, (C10×Dic3)⋊7C2, (C2×Dic3)⋊1C10, C22.6(S3×C10), C10.22(C3⋊D4), (C2×C30).45C22, (C2×C4)⋊1(C5×S3), C31(C5×C22⋊C4), (S3×C2×C10).3C2, C2.2(C5×C3⋊D4), (C2×C6).6(C2×C10), SmallGroup(240,59)

Series: Derived Chief Lower central Upper central

C1C6 — C5×D6⋊C4
C1C3C6C2×C6C2×C30S3×C2×C10 — C5×D6⋊C4
C3C6 — C5×D6⋊C4
C1C2×C10C2×C20

Generators and relations for C5×D6⋊C4
 G = < a,b,c,d | a5=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 152 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4, C23, C10 [×3], C10 [×2], Dic3, C12, D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4, C20 [×2], C2×C10, C2×C10 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], C2×C20, C2×C20, C22×C10, D6⋊C4, C5×Dic3, C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C5×C22⋊C4, C10×Dic3, C2×C60, S3×C2×C10, C5×D6⋊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, C2×C20, C5×D4 [×2], D6⋊C4, S3×C10, C5×C22⋊C4, S3×C20, C5×D12, C5×C3⋊D4, C5×D6⋊C4

Smallest permutation representation of C5×D6⋊C4
On 120 points
Generators in S120
(1 53 41 29 17)(2 54 42 30 18)(3 49 37 25 13)(4 50 38 26 14)(5 51 39 27 15)(6 52 40 28 16)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(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 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 14)(15 18)(16 17)(19 20)(21 24)(22 23)(25 26)(27 30)(28 29)(31 32)(33 36)(34 35)(37 38)(39 42)(40 41)(43 44)(45 48)(46 47)(49 50)(51 54)(52 53)(55 56)(57 60)(58 59)(61 65)(62 64)(67 71)(68 70)(73 77)(74 76)(79 83)(80 82)(85 89)(86 88)(91 95)(92 94)(97 101)(98 100)(103 107)(104 106)(109 113)(110 112)(115 119)(116 118)
(1 71 11 65)(2 72 12 66)(3 67 7 61)(4 68 8 62)(5 69 9 63)(6 70 10 64)(13 79 19 73)(14 80 20 74)(15 81 21 75)(16 82 22 76)(17 83 23 77)(18 84 24 78)(25 91 31 85)(26 92 32 86)(27 93 33 87)(28 94 34 88)(29 95 35 89)(30 96 36 90)(37 103 43 97)(38 104 44 98)(39 105 45 99)(40 106 46 100)(41 107 47 101)(42 108 48 102)(49 115 55 109)(50 116 56 110)(51 117 57 111)(52 118 58 112)(53 119 59 113)(54 120 60 114)

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

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

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

C5×D6⋊C4 is a maximal subgroup of
(C2×C20).D6  D6⋊C4.D5  C605C4⋊C2  Dic5.8D12  D30.35D4  D6⋊Dic5.C2  C5⋊(C423S3)  D6.(C4×D5)  (S3×Dic5)⋊C4  Dic54D12  Dic1514D4  Dic15⋊D4  D61Dic10  D10.17D12  D62Dic10  D30⋊D4  D6.D20  D63Dic10  D64Dic10  D30.7D4  D6⋊(C4×D5)  Dic159D4  D6⋊C4⋊D5  D10⋊D12  D6.9D20  D3012D4  Dic15.10D4  Dic15.31D4  D30.27D4  D64D20  D305D4  C20×D12  C5×S3×C22⋊C4  C20×C3⋊D4

90 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B5C5D6A6B6C10A···10L10M···10T12A12B12C12D15A15B15C15D20A···20H20I···20P30A···30L60A···60P
order12222234444555566610···1010···10121212121515151520···2020···2030···3060···60
size1111662226611112221···16···6222222222···26···62···22···2

90 irreducible representations

dim1111111111222222222222
type++++++++
imageC1C2C2C2C4C5C10C10C10C20S3D4D6C4×S3D12C3⋊D4C5×S3C5×D4S3×C10S3×C20C5×D12C5×C3⋊D4
kernelC5×D6⋊C4C10×Dic3C2×C60S3×C2×C10S3×C10D6⋊C4C2×Dic3C2×C12C22×S3D6C2×C20C30C2×C10C10C10C10C2×C4C6C22C2C2C2
# reps11114444416121222484888

Matrix representation of C5×D6⋊C4 in GL4(𝔽61) generated by

9000
0900
0090
0009
,
60000
06000
00060
0011
,
60000
23100
00060
00600
,
535500
21800
005243
00189
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,60,1],[60,23,0,0,0,1,0,0,0,0,0,60,0,0,60,0],[53,21,0,0,55,8,0,0,0,0,52,18,0,0,43,9] >;

C5×D6⋊C4 in GAP, Magma, Sage, TeX

C_5\times D_6\rtimes C_4
% in TeX

G:=Group("C5xD6:C4");
// GroupNames label

G:=SmallGroup(240,59);
// by ID

G=gap.SmallGroup(240,59);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,505,127,5765]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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