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G = C15xC4wrC2order 480 = 25·3·5

Direct product of C15 and C4wrC2

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

Aliases: C15xC4wrC2, D4:2C60, Q8:3C60, C42:6C30, C60.251D4, M4(2):4C30, (C4xC20):16C6, (C4xC60):24C2, (C3xD4):5C20, (C5xD4):8C12, C4.3(C2xC60), (C3xQ8):5C20, (C4xC12):10C10, (D4xC15):17C4, (C5xQ8):11C12, (Q8xC15):17C4, C4oD4.3C30, C12.66(C5xD4), C4.17(D4xC15), C20.66(C3xD4), C20.52(C2xC12), C60.227(C2xC4), C12.31(C2xC20), (C2xC30).129D4, C22.3(D4xC15), (C5xM4(2)):10C6, (C3xM4(2)):10C10, (C15xM4(2)):22C2, (C2xC60).571C22, C30.130(C22:C4), (C5xC4oD4).8C6, (C2xC6).22(C5xD4), (C2xC4).18(C2xC30), (C3xC4oD4).4C10, (C2xC10).23(C3xD4), C2.8(C15xC22:C4), C6.26(C5xC22:C4), (C2xC20).119(C2xC6), (C15xC4oD4).10C2, C10.37(C3xC22:C4), (C2xC12).122(C2xC10), SmallGroup(480,207)

Series: Derived Chief Lower central Upper central

C1C4 — C15xC4wrC2
C1C2C4C2xC4C2xC20C2xC60C15xM4(2) — C15xC4wrC2
C1C2C4 — C15xC4wrC2
C1C60C2xC60 — C15xC4wrC2

Generators and relations for C15xC4wrC2
 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, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C10, C10, C12, C12, C2xC6, C2xC6, C15, C42, M4(2), C4oD4, C20, C20, C2xC10, C2xC10, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C30, C30, C4wrC2, C40, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C4xC12, C3xM4(2), C3xC4oD4, C60, C60, C2xC30, C2xC30, C4xC20, C5xM4(2), C5xC4oD4, C3xC4wrC2, C120, C2xC60, C2xC60, D4xC15, D4xC15, Q8xC15, C5xC4wrC2, C4xC60, C15xM4(2), C15xC4oD4, C15xC4wrC2
Quotients: C1, C2, C3, C4, C22, C5, C6, C2xC4, D4, C10, C12, C2xC6, C15, C22:C4, C20, C2xC10, C2xC12, C3xD4, C30, C4wrC2, C2xC20, C5xD4, C3xC22:C4, C60, C2xC30, C5xC22:C4, C3xC4wrC2, C2xC60, D4xC15, C5xC4wrC2, C15xC22:C4, C15xC4wrC2

Smallest permutation representation of C15xC4wrC2
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 67 55 21)(2 68 56 22)(3 69 57 23)(4 70 58 24)(5 71 59 25)(6 72 60 26)(7 73 46 27)(8 74 47 28)(9 75 48 29)(10 61 49 30)(11 62 50 16)(12 63 51 17)(13 64 52 18)(14 65 53 19)(15 66 54 20)(31 96 120 87)(32 97 106 88)(33 98 107 89)(34 99 108 90)(35 100 109 76)(36 101 110 77)(37 102 111 78)(38 103 112 79)(39 104 113 80)(40 105 114 81)(41 91 115 82)(42 92 116 83)(43 93 117 84)(44 94 118 85)(45 95 119 86)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 106)(13 107)(14 108)(15 109)(16 87)(17 88)(18 89)(19 90)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 46)(43 47)(44 48)(45 49)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 91)(73 92)(74 93)(75 94)
(1 21 55 67)(2 22 56 68)(3 23 57 69)(4 24 58 70)(5 25 59 71)(6 26 60 72)(7 27 46 73)(8 28 47 74)(9 29 48 75)(10 30 49 61)(11 16 50 62)(12 17 51 63)(13 18 52 64)(14 19 53 65)(15 20 54 66)

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

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

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

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++++++
imageC1C2C2C2C3C4C4C5C6C6C6C10C10C10C12C12C15C20C20C30C30C30C60C60D4D4C3xD4C3xD4C4wrC2C5xD4C5xD4C3xC4wrC2D4xC15D4xC15C5xC4wrC2C15xC4wrC2
kernelC15xC4wrC2C4xC60C15xM4(2)C15xC4oD4C5xC4wrC2D4xC15Q8xC15C3xC4wrC2C4xC20C5xM4(2)C5xC4oD4C4xC12C3xM4(2)C3xC4oD4C5xD4C5xQ8C4wrC2C3xD4C3xQ8C42M4(2)C4oD4D4Q8C60C2xC30C20C2xC10C15C12C2xC6C5C4C22C3C1
# reps1111222422244444888888161611224448881632

Matrix representation of C15xC4wrC2 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] >;

C15xC4wrC2 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

׿
x
:
Z
F
o
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