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G = D6016C4order 480 = 25·3·5

10th semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6016C4, C60.101D4, C20.57D12, Dic3016C4, C1510C4≀C2, C20.41(C4×S3), (C4×Dic5)⋊2S3, (C2×C20).57D6, C12.33(C4×D5), (C2×C6).33D20, (C2×C30).28D4, C4.Dic31D5, C53(C424S3), C31(D207C4), C60.110(C2×C4), (C12×Dic5)⋊2C2, C10.24(D6⋊C4), (C2×C12).313D10, C4.29(C5⋊D12), C12.65(C5⋊D4), (C2×C60).44C22, C4.8(D30.C2), D6011C2.6C2, C6.9(D10⋊C4), C30.64(C22⋊C4), C22.2(C3⋊D20), C2.10(D304C4), (C2×C4).138(S3×D5), (C5×C4.Dic3)⋊4C2, (C2×C10).3(C3⋊D4), SmallGroup(480,57)

Series: Derived Chief Lower central Upper central

C1C60 — D6016C4
C1C5C15C30C2×C30C2×C60C12×Dic5 — D6016C4
C15C30C60 — D6016C4
C1C4C2×C4

Generators and relations for D6016C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a49, cbc-1=a3b >

Subgroups: 524 in 88 conjugacy classes, 32 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, D5, C10, C10, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, D15, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, C4×C12, C4○D12, C3×Dic5 [×2], Dic15, C60 [×2], D30, C2×C30, C4×Dic5, C5×M4(2), C4○D20, C424S3, C5×C3⋊C8, C6×Dic5, Dic30, C4×D15, D60, C157D4, C2×C60, D207C4, C12×Dic5, C5×C4.Dic3, D6011C2, D6016C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4≀C2, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C424S3, D30.C2, C3⋊D20, C5⋊D12, D207C4, D304C4, D6016C4

Smallest permutation representation of D6016C4
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 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 84)(12 83)(13 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)
(1 31)(2 20)(3 9)(4 58)(5 47)(6 36)(7 25)(8 14)(10 52)(11 41)(12 30)(13 19)(15 57)(16 46)(17 35)(18 24)(21 51)(22 40)(23 29)(26 56)(27 45)(28 34)(32 50)(33 39)(37 55)(38 44)(42 60)(43 49)(48 54)(53 59)(61 94 91 64)(62 83 92 113)(63 72 93 102)(65 110 95 80)(66 99 96 69)(67 88 97 118)(68 77 98 107)(70 115 100 85)(71 104 101 74)(73 82 103 112)(75 120 105 90)(76 109 106 79)(78 87 108 117)(81 114 111 84)(86 119 116 89)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B6C8A8B10A10B10C10D12A12B12C12D12E···12L15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order12223444444445566688101010101212121212···12151520202020202030···3040···4060···60
size11260211210101010602222212122244222210···10442222444···412···124···4

66 irreducible representations

dim11111122222222222222444444
type++++++++++++++++
imageC1C2C2C2C4C4S3D4D4D5D6D10C4×S3D12C3⋊D4C4≀C2C4×D5C5⋊D4D20C424S3S3×D5D30.C2C5⋊D12C3⋊D20D207C4D6016C4
kernelD6016C4C12×Dic5C5×C4.Dic3D6011C2Dic30D60C4×Dic5C60C2×C30C4.Dic3C2×C20C2×C12C20C20C2×C10C15C12C12C2×C6C5C2×C4C4C4C22C3C1
# reps11112211121222244448222248

Matrix representation of D6016C4 in GL4(𝔽241) generated by

181000
144400
0019051
00190240
,
2117000
12522000
000240
002400
,
240000
3917700
0051190
00240190
G:=sub<GL(4,GF(241))| [181,144,0,0,0,4,0,0,0,0,190,190,0,0,51,240],[21,125,0,0,170,220,0,0,0,0,0,240,0,0,240,0],[240,39,0,0,0,177,0,0,0,0,51,240,0,0,190,190] >;

D6016C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{16}C_4
% in TeX

G:=Group("D60:16C4");
// GroupNames label

G:=SmallGroup(480,57);
// by ID

G=gap.SmallGroup(480,57);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,219,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^3*b>;
// generators/relations

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