Copied to
clipboard

G = C15×C4≀C2order 480 = 25·3·5

Direct product of C15 and C4≀C2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C15×C4≀C2, D42C60, Q83C60, C426C30, C60.251D4, M4(2)⋊4C30, (C4×C20)⋊16C6, (C4×C60)⋊24C2, (C3×D4)⋊5C20, (C5×D4)⋊8C12, C4.3(C2×C60), (C3×Q8)⋊5C20, (C4×C12)⋊10C10, (D4×C15)⋊17C4, (C5×Q8)⋊11C12, (Q8×C15)⋊17C4, C4○D4.3C30, C12.66(C5×D4), C4.17(D4×C15), C20.66(C3×D4), C20.52(C2×C12), C60.227(C2×C4), C12.31(C2×C20), (C2×C30).129D4, C22.3(D4×C15), (C5×M4(2))⋊10C6, (C3×M4(2))⋊10C10, (C15×M4(2))⋊22C2, (C2×C60).571C22, C30.130(C22⋊C4), (C5×C4○D4).8C6, (C2×C6).22(C5×D4), (C2×C4).18(C2×C30), (C3×C4○D4).4C10, (C2×C10).23(C3×D4), C2.8(C15×C22⋊C4), C6.26(C5×C22⋊C4), (C2×C20).119(C2×C6), (C15×C4○D4).10C2, C10.37(C3×C22⋊C4), (C2×C12).122(C2×C10), SmallGroup(480,207)

Series: Derived Chief Lower central Upper central

C1C4 — C15×C4≀C2
C1C2C4C2×C4C2×C20C2×C60C15×M4(2) — C15×C4≀C2
C1C2C4 — C15×C4≀C2
C1C60C2×C60 — C15×C4≀C2

Generators and relations for C15×C4≀C2
 G = < a,b,c,d | a15=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 136 in 88 conjugacy classes, 48 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4, D4, Q8, C10, C10 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6, C15, C42, M4(2), C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C24, C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C30, C30 [×2], C4≀C2, C40, C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C4×C12, C3×M4(2), C3×C4○D4, C60 [×2], C60 [×3], C2×C30, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, C3×C4≀C2, C120, C2×C60, C2×C60 [×2], D4×C15, D4×C15, Q8×C15, C5×C4≀C2, C4×C60, C15×M4(2), C15×C4○D4, C15×C4≀C2
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4 [×2], C10 [×3], C12 [×2], C2×C6, C15, C22⋊C4, C20 [×2], C2×C10, C2×C12, C3×D4 [×2], C30 [×3], C4≀C2, C2×C20, C5×D4 [×2], C3×C22⋊C4, C60 [×2], C2×C30, C5×C22⋊C4, C3×C4≀C2, C2×C60, D4×C15 [×2], C5×C4≀C2, C15×C22⋊C4, C15×C4≀C2

Smallest permutation representation of C15×C4≀C2
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 31 108 105)(2 32 109 91)(3 33 110 92)(4 34 111 93)(5 35 112 94)(6 36 113 95)(7 37 114 96)(8 38 115 97)(9 39 116 98)(10 40 117 99)(11 41 118 100)(12 42 119 101)(13 43 120 102)(14 44 106 103)(15 45 107 104)(16 49 78 64)(17 50 79 65)(18 51 80 66)(19 52 81 67)(20 53 82 68)(21 54 83 69)(22 55 84 70)(23 56 85 71)(24 57 86 72)(25 58 87 73)(26 59 88 74)(27 60 89 75)(28 46 90 61)(29 47 76 62)(30 48 77 63)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 46)(15 47)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 31)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 104)(77 105)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)
(1 105 108 31)(2 91 109 32)(3 92 110 33)(4 93 111 34)(5 94 112 35)(6 95 113 36)(7 96 114 37)(8 97 115 38)(9 98 116 39)(10 99 117 40)(11 100 118 41)(12 101 119 42)(13 102 120 43)(14 103 106 44)(15 104 107 45)

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

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

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

210 conjugacy classes

class 1 2A2B2C3A3B4A4B4C···4G4H5A5B5C5D6A6B6C6D6E6F8A8B10A10B10C10D10E10F10G10H10I10J10K10L12A12B12C12D12E···12N12O12P15A···15H20A···20H20I···20AB20AC20AD20AE20AF24A24B24C24D30A···30H30I···30P30Q···30X40A···40H60A···60P60Q···60BD60BE···60BL120A···120P
order122233444···445555666666881010101010101010101010101212121212···12121215···1520···2020···20202020202424242430···3030···3030···3040···4060···6060···6060···60120···120
size112411112···2411111122444411112222444411112···2441···11···12···2444444441···12···24···44···41···12···24···44···4

210 irreducible representations

dim111111111111111111111111222222222222
type++++++
imageC1C2C2C2C3C4C4C5C6C6C6C10C10C10C12C12C15C20C20C30C30C30C60C60D4D4C3×D4C3×D4C4≀C2C5×D4C5×D4C3×C4≀C2D4×C15D4×C15C5×C4≀C2C15×C4≀C2
kernelC15×C4≀C2C4×C60C15×M4(2)C15×C4○D4C5×C4≀C2D4×C15Q8×C15C3×C4≀C2C4×C20C5×M4(2)C5×C4○D4C4×C12C3×M4(2)C3×C4○D4C5×D4C5×Q8C4≀C2C3×D4C3×Q8C42M4(2)C4○D4D4Q8C60C2×C30C20C2×C10C15C12C2×C6C5C4C22C3C1
# reps1111222422244444888888161611224448881632

Matrix representation of C15×C4≀C2 in GL2(𝔽241) generated by

940
094
,
640
152177
,
89128
149152
,
1770
2091
G:=sub<GL(2,GF(241))| [94,0,0,94],[64,152,0,177],[89,149,128,152],[177,209,0,1] >;

C15×C4≀C2 in GAP, Magma, Sage, TeX

C_{15}\times C_4\wr C_2
% in TeX

G:=Group("C15xC4wrC2");
// GroupNames label

G:=SmallGroup(480,207);
// by ID

G=gap.SmallGroup(480,207);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,10504,5261,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^4=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^-1*c>;
// generators/relations

׿
×
𝔽